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R<sub>XE</sub>/19E is a X8-digit prime (XE is the only known prime ''p'' such that R''<sub>p</sub>''/(2''p''+1) is prime). |
R<sub>XE</sub>/19E is a X8-digit prime (XE is the only known prime ''p'' such that R''<sub>p</sub>''/(2''p''+1) is prime). |
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− | The largest two prime factors of R<sub>141</sub> both have 60 digits, and these two prime factors are very close. (they are 10E6370EE18220X59650958X71279E43117722E6XE34EX80648E2256241EX2105E527461 and 11424383274E97427074X9354XX73238X2661601841X7805E629766262X78X928600X46E, and the product of them is an EE-digit number, 1250268E4463424X802E319385467207EE6752954357X1339XE38X210408945194E2EE1285195481E272114954389086E076XE430X178293097E5388067X829X2X09E209737X60E) |
+ | The largest two prime factors of R<sub>141</sub> both have 60 (half of 100) digits, and these two prime factors are very close. (they are 10E6370EE18220X59650958X71279E43117722E6XE34EX80648E2256241EX2105E527461 and 11424383274E97427074X9354XX73238X2661601841X7805E629766262X78X928600X46E, and the product of them is an EE-digit number, 1250268E4463424X802E319385467207EE6752954357X1339XE38X210408945194E2EE1285195481E272114954389086E076XE430X178293097E5388067X829X2X09E209737X60E) |
If ''n'' is composite, then R''<sub>n</sub>'' is also composite (e.g. 2E = 5 × 7, and R<sub>2E</sub> = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001), however, when ''n'' is prime, R''<sub>n</sub>'' may not be prime, the first example is ''n''=7, although 7 is prime, R<sub>7</sub> = 1111111 is not prime, it equals 46E × 2X3E. |
If ''n'' is composite, then R''<sub>n</sub>'' is also composite (e.g. 2E = 5 × 7, and R<sub>2E</sub> = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001), however, when ''n'' is prime, R''<sub>n</sub>'' may not be prime, the first example is ''n''=7, although 7 is prime, R<sub>7</sub> = 1111111 is not prime, it equals 46E × 2X3E. |
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For prime ''p'' other than 2, 3 and E, the smallest integer ''n'' ≥ 1 such that ''p'' divides ''R<sub>n</sub>'' is the period length of 1/''p'', e.g. none of 1, 11, 111, 1111 and 11111 is divisible by 17, but 111111 is, and the period length of 1/17 is 6: 1/17 = 0.<span style="text-decoration:overline;">076E45</span> |
For prime ''p'' other than 2, 3 and E, the smallest integer ''n'' ≥ 1 such that ''p'' divides ''R<sub>n</sub>'' is the period length of 1/''p'', e.g. none of 1, 11, 111, 1111 and 11111 is divisible by 17, but 111111 is, and the period length of 1/17 is 6: 1/17 = 0.<span style="text-decoration:overline;">076E45</span> |
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+ | |||
+ | All numbers ''n'' coprime to 10 has multiple which is repunit, if ''n'' is coprime to 10, the repunit ''R''<sub>''phi''(E''n'')</sub> (where ''phi'' is Euler totient function) is always divisible by ''n'' |
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All repunit composites with prime length except ''R''<sub>E</sub> are Fermat pseudoprime (also Euler pseudoprime, Euler-Jacobi pseudoprime and strong pseudoprime) base 10, and hence deceptive primes. |
All repunit composites with prime length except ''R''<sub>E</sub> are Fermat pseudoprime (also Euler pseudoprime, Euler-Jacobi pseudoprime and strong pseudoprime) base 10, and hence deceptive primes. |
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Line 636: | Line 638: | ||
** bases of the form 4''k''<sup>4</sup> with no repunit primes: 54, 230, 714, 1544, 3000, 5684, 9594, 13230, 1E194, 29X84, 40000, 56144, 74E14, 99230, 107854, 141404, 183000, 211804, 26X454, 316230, 392314, 45E944, 540000, 634284, 741994, 866230, 9X6994, E45284, 1103000, 12X1944, 14X3314, 1709230, 1959454, 2015804, 2300000, 2616404, 2962854, 3123230, 3519E14, 3951144, 4203000, 46E5X84, 5030194, 55E0230, 5EE8594, 6653684, 7140000, 7880544, 8457714, 9090230, 9965054, X6X1004, E483000, 10312004, 11211054, 12183230, 131XE714, 14295544, 15440000, 1666X684, 17964594, 19129230, 1X584194, 1EXE4X84, 21503000, 22EE2144, 24785E14, 26446230, 281E6854, 2X05E404, 30000000, 32060804, 34205454, 36476230, 38837314, 3E110944, 41703000, 44216284, 46X52994, 497E9230, 50695994, 536X9284, 56840000, 59E16944, 6133X314, 648E3230, 68402454, 70070804, 73X83000, 77X42404, 7EE73854, 84260230, 88708E14, 9113E144, 95940000, 9X714X84, X3687194, X8820230, E1E61594, E7494684, 101003000, 106932544, 110868714, 1169E3230, 121154054, 127715004, 132300000, 139117004, 144168054, 14E439230, 156954714, 162500544, 16X303000, 176366684, 182675594, 18E036230, 197873194, 1X4772X84, 1E1940000, 1EE1X1144, 208920E14, 216746230, 22485E854, 233070404, 241783000, 2505X2804, 25E716454, 26XE69230, 27X726314, 28X614944, 29X840000, 2XE1XE284, 2EEXX9994, 310E43230, 32233X994, 333XX4284, 345X03000, 3580X2944, 36X74E314, 381590230, 3947E1454, 3X81EX804, 400000000, ... |
** bases of the form 4''k''<sup>4</sup> with no repunit primes: 54, 230, 714, 1544, 3000, 5684, 9594, 13230, 1E194, 29X84, 40000, 56144, 74E14, 99230, 107854, 141404, 183000, 211804, 26X454, 316230, 392314, 45E944, 540000, 634284, 741994, 866230, 9X6994, E45284, 1103000, 12X1944, 14X3314, 1709230, 1959454, 2015804, 2300000, 2616404, 2962854, 3123230, 3519E14, 3951144, 4203000, 46E5X84, 5030194, 55E0230, 5EE8594, 6653684, 7140000, 7880544, 8457714, 9090230, 9965054, X6X1004, E483000, 10312004, 11211054, 12183230, 131XE714, 14295544, 15440000, 1666X684, 17964594, 19129230, 1X584194, 1EXE4X84, 21503000, 22EE2144, 24785E14, 26446230, 281E6854, 2X05E404, 30000000, 32060804, 34205454, 36476230, 38837314, 3E110944, 41703000, 44216284, 46X52994, 497E9230, 50695994, 536X9284, 56840000, 59E16944, 6133X314, 648E3230, 68402454, 70070804, 73X83000, 77X42404, 7EE73854, 84260230, 88708E14, 9113E144, 95940000, 9X714X84, X3687194, X8820230, E1E61594, E7494684, 101003000, 106932544, 110868714, 1169E3230, 121154054, 127715004, 132300000, 139117004, 144168054, 14E439230, 156954714, 162500544, 16X303000, 176366684, 182675594, 18E036230, 197873194, 1X4772X84, 1E1940000, 1EE1X1144, 208920E14, 216746230, 22485E854, 233070404, 241783000, 2505X2804, 25E716454, 26XE69230, 27X726314, 28X614944, 29X840000, 2XE1XE284, 2EEXX9994, 310E43230, 32233X994, 333XX4284, 345X03000, 3580X2944, 36X74E314, 381590230, 3947E1454, 3X81EX804, 400000000, ... |
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− | == List of repunit primes base b == |
+ | == List of repunit (probable) primes base b == |
We also consider “negative primes” (e.g. R<sub>2</sub>(−10) = −E) as primes, since they are primes in the domain Z (the set of all integers). |
We also consider “negative primes” (e.g. R<sub>2</sub>(−10) = −E) as primes, since they are primes in the domain Z (the set of all integers). |
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{|class="wikitable" |
{|class="wikitable" |
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!''b'' |
!''b'' |
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− | ! |
+ | !length of repunit (probable) primes in base ''b'' |
+ | |- |
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+ | |−100 |
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+ | |3, 1E, 35, 225, 1E4E, 22237, 59307, ... |
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+ | |- |
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+ | |−EE |
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+ | |7, 15, 17, 3E, 87, 2687, X6EE, ... |
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+ | |- |
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+ | |−EX |
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+ | |3, 4441, ... |
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+ | |- |
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+ | |−E9 |
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+ | |5, X27, ... |
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+ | |- |
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+ | |−E8 |
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+ | |2, 4E, 1530E, ... |
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+ | |- |
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+ | |−E7 |
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+ | |3, 15, 3E, 1677, 16X7, 13371, ... |
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+ | |- |
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+ | |−E6 |
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+ | |2, 87, 401, 62X5, ... |
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+ | |- |
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+ | |−E5 |
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+ | |85, 181, 255, 11X7, 1078E, ... |
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+ | |- |
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+ | |−E4 |
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+ | |5, 7, 1E, 4E, 147, 1231, 3617, ... |
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+ | |- |
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+ | |−E3 |
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+ | |5, 7, 1667, 6E01, ... |
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+ | |- |
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+ | |−E2 |
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+ | |11, 817, 3X91, ... |
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+ | |- |
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+ | |−E1 |
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+ | |5, 7, 15, 4E, 67, 111, ... |
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+ | |- |
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+ | |−E0 |
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+ | |2, 3, 85, 111, 907, ... |
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+ | |- |
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+ | |−XE |
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+ | |5, 85, 1E65, 20X5, ... |
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+ | |- |
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+ | |−XX |
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+ | |32E, ... |
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+ | |- |
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+ | |−X9 |
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+ | |15, 16E, 1021, ... |
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+ | |- |
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+ | |−X8 |
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+ | |2, 7 (no others) |
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+ | |- |
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+ | |−X7 |
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+ | |225, 745, 119X7, ... |
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+ | |- |
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+ | |−X6 |
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+ | |5, 11, 3E, 117, 17E, 274E, ... |
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+ | |- |
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+ | |−X5 |
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+ | |(none) |
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+ | |- |
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+ | |−X4 |
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+ | |960E, ... |
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+ | |- |
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+ | |−X3 |
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+ | |25, 517, ... |
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+ | |- |
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+ | |−X2 |
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+ | |205, 22E1, 7561, 10X85, ... |
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+ | |- |
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+ | |−X1 |
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+ | |5, 11, 81, X4E, 6675, 13556E, ... |
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|- |
|- |
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|−X0 |
|−X0 |
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Line 700: | Line 774: | ||
|- |
|- |
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|−85 |
|−85 |
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− | |7, 171, 44E1E, ... |
+ | |7, 171, 44E1E, 80681, ... |
|- |
|- |
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|−84 |
|−84 |
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Line 709: | Line 783: | ||
|- |
|- |
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|−82 |
|−82 |
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− | |2, 17, 85, 39725, 56577, ... |
+ | |2, 17, 85, 39725, 56577, 91897, ... |
|- |
|- |
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|−81 |
|−81 |
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Line 739: | Line 813: | ||
|- |
|- |
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|−74 |
|−74 |
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− | |4E1, 965, 2E89E, ... |
+ | |4E1, 965, 2E89E, X096E, ... |
|- |
|- |
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|−73 |
|−73 |
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Line 763: | Line 837: | ||
|- |
|- |
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|−68 |
|−68 |
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− | |2, 5, 11, 16E, 307, ... |
+ | |2, 5, 11, 16E, 307, 93101, 93231, E022E, ... |
|- |
|- |
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|−67 |
|−67 |
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Line 775: | Line 849: | ||
|- |
|- |
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|−64 |
|−64 |
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− | |3, 5, 13E, 1X5, 11771, ... |
+ | |3, 5, 13E, 1X5, 11771, 8004E, ... |
|- |
|- |
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|−63 |
|−63 |
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Line 781: | Line 855: | ||
|- |
|- |
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|−62 |
|−62 |
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− | |2, 11, 27, 31, 91, X087, ... |
+ | |2, 11, 27, 31, 91, X087, 809X7, ... |
|- |
|- |
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|−61 |
|−61 |
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Line 814: | Line 888: | ||
|- |
|- |
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|−53 |
|−53 |
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− | |3, 31, 35, 1297, 23E7, 10X8E, 25927, 4E107, ... |
+ | |3, 31, 35, 1297, 23E7, 10X8E, 25927, 4E107, 90X6E, E6015, ... |
|- |
|- |
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|−52 |
|−52 |
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− | |2, E, 25, 11E, 221, 9707, 1X48E, ... |
+ | |2, E, 25, 11E, 221, 9707, 1X48E, 82837, ... |
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|- |
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|−51 |
|−51 |
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Line 904: | Line 978: | ||
|- |
|- |
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|−29 |
|−29 |
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− | |5, 57, 111, 7097, ... |
+ | |5, 57, 111, 7097, 131555, ... |
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|- |
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|−28 |
|−28 |
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Line 919: | Line 993: | ||
|- |
|- |
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|−24 |
|−24 |
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− | |3, 17, 271, 2XE, 34E, 71E, 403X1, ... |
+ | |3, 17, 271, 2XE, 34E, 71E, 403X1, X9331, ... |
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|- |
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|−23 |
|−23 |
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Line 970: | Line 1,044: | ||
|- |
|- |
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|−E |
|−E |
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− | |5, 7, 12E, 171, 307, 3X5, 3655, X9767, 1392X1, ... |
+ | |5, 7, 12E, 171, 307, 3X5, 3655, X9767, 1392X1, 912657, ... |
|- |
|- |
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|−X |
|−X |
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Line 976: | Line 1,050: | ||
|- |
|- |
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|−9 |
|−9 |
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− | |3, 4E, 167, 397, 545, 701, 107E, 224E, 2459E, 939EE, 29E081, ... |
+ | |3, 4E, 167, 397, 545, 701, 107E, 224E, 2459E, 939EE, 29E081, 355851, ... |
|- |
|- |
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|−8 |
|−8 |
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Line 1,024: | Line 1,098: | ||
|- |
|- |
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|X |
|X |
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− | |2, 17, 1E, 225, 71E, 244X1, 42045, 53301, 110547, ... |
+ | |2, 17, 1E, 225, 71E, 244X1, 42045, 53301, 110547, 1E35561, 28X421E, ... |
|- |
|- |
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|E |
|E |
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− | |15, 17, 61, E7, 637, 112E, 1211, 2941, 2E95, 6357, E801, 12205E, ... |
+ | |15, 17, 61, E7, 637, 112E, 1211, 2941, 2E95, 6357, E801, 12205E, 761707, ... |
|- |
|- |
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|10 |
|10 |
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Line 1,165: | Line 1,239: | ||
|- |
|- |
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|49 |
|49 |
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− | |3, 15, 91, 107, 157, 471, 9997, 10905, ... |
+ | |3, 15, 91, 107, 157, 471, 9997, 10905, 113645, ... |
|- |
|- |
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|4X |
|4X |
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Line 1,228: | Line 1,302: | ||
|- |
|- |
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|66 |
|66 |
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− | |2, 3, 85, 195, 1165, 32X31, ... |
+ | |2, 3, 85, 195, 1165, 32X31, E9X7E, ... |
|- |
|- |
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|67 |
|67 |
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Line 1,255: | Line 1,329: | ||
|- |
|- |
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|73 |
|73 |
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− | |7, 15, 5X37E, ... |
+ | |7, 15, 5X37E, 105161, ... |
|- |
|- |
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|74 |
|74 |
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Line 1,970: | Line 2,044: | ||
== The generalized repunit conjecture == |
== The generalized repunit conjecture == |
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− | A conjecture related to the generalized repunit primes:<ref>[http://primes.utm.edu/mersenne/heuristic.html Deriving the Wagstaff Mersenne Conjecture]</ref><ref>[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No+Match%3BMatch%3BMatches&z=4 Generalized Repunit Conjecture]</ref> (the conjecture predicts where is the next generalized |
+ | A conjecture related to the generalized repunit primes:<ref>[http://primes.utm.edu/mersenne/heuristic.html Deriving the Wagstaff Mersenne Conjecture]</ref><ref>[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No+Match%3BMatch%3BMatches&z=4 Generalized Repunit Conjecture]</ref> (the conjecture predicts where is the next generalized repunit prime, if the conjecture is true, then there are infinitely many repunit primes for all bases <math>b</math>) For any integer <math>b</math>, which satisfies the conditions: |
# <math>|b|>1</math>. |
# <math>|b|>1</math>. |
Revision as of 16:19, 21 July 2021
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit.
A repunit prime is a repunit that is also a prime number.
In the sections below, Rn is the repunit with length n, e.g. R10 = 111111111111.
Repunit prime
The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.
It is easy to show that if n is divisible by a, then Rn(b) is divisible by Ra(b):
where is the cyclotomic polynomial and d ranges over the divisors of n. For p prime,
which has the expected form of a repunit when x is substituted with b.
For example, 10 is divisible by 2, 3, 4, and 6, thus R10 is divisible by R2, R3, R4, and R6, in fact, 111111111111 = 11 · 10101010101 = 111 · 1001001001 = 1111 · 100010001 = 111111 · 1000001, the corresponding cyclotomic polynomials and and and and are and and and and , respectively (the algebraic factors of R10 = 111111111111 is 11 · E1 · 101 · 111 · EE01). Thus, for Rn to be prime, n must necessarily be prime, but it is not sufficient for n to be prime. For example, R7 = 1111111 = 46E · 2X3E is not prime. Except for the case of RE = 11111111111 = E · 1E · 754E2E41, p can only divide Rn for prime n if p = 2kn + 1 for some k.
Rn is known to be prime for n = 2, 3, 5, 17, 81, 91, 225, 255, 4X5, and Rn is probable prime for n = 5777, 879E, 198E1, 23175, 311407. (Note that 879E is the only known such n ends with E)
RXE/19E is a X8-digit prime (XE is the only known prime p such that Rp/(2p+1) is prime).
The largest two prime factors of R141 both have 60 (half of 100) digits, and these two prime factors are very close. (they are 10E6370EE18220X59650958X71279E43117722E6XE34EX80648E2256241EX2105E527461 and 11424383274E97427074X9354XX73238X2661601841X7805E629766262X78X928600X46E, and the product of them is an EE-digit number, 1250268E4463424X802E319385467207EE6752954357X1339XE38X210408945194E2EE1285195481E272114954389086E076XE430X178293097E5388067X829X2X09E209737X60E)
If n is composite, then Rn is also composite (e.g. 2E = 5 × 7, and R2E = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001), however, when n is prime, Rn may not be prime, the first example is n=7, although 7 is prime, R7 = 1111111 is not prime, it equals 46E × 2X3E.
The repunit prime 111 is exactly the largest known negative base −n such that the repunit with length n is prime. (for positive base, the largest known such n is 444E, note that 444E is a near-repdigit prime)
Theorem
If p is prime other than E, then every prime factor of Rp is = 1 mod p. (e.g. R7 = 46E × 2X3E, and both 46E and 2X3E are = 1 mod 7)
If p is Sophie Germain prime other than 2, 3 and 5, then Rp is composite, since Rp must be divisible by 2p+1. (e.g. 1E|RE, 3E|R1E, 4E|R25, 6E|R35, 8E|R45, 11E|R6E, 12E|R75, 16E|R95, 19E|RXE) (by the way, R2, R3 and R5 are all primes)
If p is prime other than 2, 3 and E, then p divides Rp-1, however, some composite numbers c also divide Rc−1, the first such example is 55, which divides R54, such composites are called deceptive primes.
For prime p other than 2, 3 and E, the smallest integer n ≥ 1 such that p divides Rn is the period length of 1/p, e.g. none of 1, 11, 111, 1111 and 11111 is divisible by 17, but 111111 is, and the period length of 1/17 is 6: 1/17 = 0.076E45
All numbers n coprime to 10 has multiple which is repunit, if n is coprime to 10, the repunit Rphi(En) (where phi is Euler totient function) is always divisible by n
All repunit composites with prime length except RE are Fermat pseudoprime (also Euler pseudoprime, Euler-Jacobi pseudoprime and strong pseudoprime) base 10, and hence deceptive primes.
All composite factors of repunits with prime length except RE are Fermat pseudoprime (also Euler pseudoprime, Euler-Jacobi pseudoprime and strong pseudoprime) base 10, and hence deceptive primes.
If n is Fermat pseudoprime base 10, then Rn is also Fermat pseudoprime base 10 (thus, there are infinitely many Fermat pseudoprimes base 10). (e.g. R55 and R77 are Fermat pseudoprime base 10)
If n is deceptive prime, then Rn is also deceptive prime (thus, there are infinitely many deceptive primes). (e.g. R55 and R77 are deceptive primes)
- If (and only if) n is divisible by 2, then Rn is divisible by 11.
- If (and only if) n is divisible by 3, then Rn is divisible by 111.
- If (and only if) n is divisible by 4, then Rn is divisible by 5 and 25.
- If (and only if) n is divisible by 5, then Rn is divisible by 11111.
- If (and only if) n is divisible by 6, then Rn is divisible by 7 and 17.
- If (and only if) n is divisible by 7, then Rn is divisible by 46E and 2X3E.
- If (and only if) n is divisible by 8, then Rn is divisible by 75 and 175.
- If (and only if) n is divisible by 9, then Rn is divisible by 31 and 3X891.
- If (and only if) n is divisible by X, then Rn is divisible by E0E1.
- If (and only if) n is divisible by E, then Rn is divisible by E, 1E, and 754E2E41.
- If (and only if) n is divisible by 10, then Rn is divisible by EE01.
- Let p be a prime >3, m is the smallest integer ≥1 such that Rm is divisible by p, then if (and only if) n is divisible by m, then Rn is divisible by p.
- Since there are no squares = 11 mod 100, thus the only repunit which is square is 1, in fact, the only repunit which is perfect power is 1.
- Rn is Harshad number if and only if n is power of E (include 1).
- If p is prime differ from 2, 3, and E, then p divides Rp−1.
- If p is prime differ from E, then p divides Rp−1.
Example of E01-type numbers
n | |||||
---|---|---|---|---|---|
R6 | R2 × R3 | 1221 | × | E1 | 7 · 17 |
RX | R2 × R5 | 122221 | E0E1 | prime | |
R12 | R2 × R7 | 12222221 | E0E0E1 | 157 · 7687 | |
R16 | R2 × R9 | 1222222221 | E0E0E0E1 | 7 · 17 · E61 · 1061 | |
R1X | R2 × RE | 122222222221 | E0E0E0E0E1 | prime |
- R10 = 11222211 × EE01
- R13 = 1233321 × E00E0EE1
- R18 = 112222222211 × EE00EE01
- R19 = 123333321 × E00E00EE0EE1
- R20 = 112233332211 × EE0000EEEE01
- R20 = 1111222222221111 × EEEE0001
- R24 = 1122222222222211 × EE00EE00EE01
- R26 = 11223333332211 × EE0000EE00EEEE01
- R26 = 111222222222222111 × EEE000EEE001
- R26 = 11111222222222211111 × EEEEE00001
- R26 = 1011121222222221211101 × 10EXXE011
- R29 = 1233333333321 × E00E00E00E0EE0EE0EE1
- R2E = 12345554321 × E0000E0E00E0E0EE0E0EEEE1
- R30 = 11222222222222222211 × EE00EE00EE00EE01
- R30 = 111222333333222111 × EEE000000EEEEEE001
- R30 = 111111222222222222111111 × EEEEEE000001
- R34 = 1122334444332211 × EE000000EEEE0000EEEEEE01
- R34 = 111122222222222222221111 × EEEE0000EEEE0001
- R36 = 112233333333332211 × EE0000EE0000EEEE00EEEE01
- R36 = 111222222222222222222111 × EEE000EEE000EEE001
- R36 = 1111111222222222222221111111 × EEEEEEE0000001
- R36 = 101111121222222222222121111101 × 10EXE00EXE011
- R38 = 112222222222222222222211 × EE00EE00EE00EE00EE01
- R39 = 1234555554321 × E0000E000EE000EE00EEE00EEE0EEEE1
- R39 = 111222333333333222111 × EEE000000EEE000EEEEEE001
- R40 = 11223333333333332211 × EE0000EE0000EE00EEEE00EEEE01
- R40 = 111122223333333322221111 × EEEE00000000EEEEEEEE0001
- R40 = 11111111222222222222222211111111 × EEEEEEEE00000001
- R4 = 11 × (E1 + 10)
- R6 = 11 × (E0E1 + 1010)
- R8 = 11 × (E0E0E1 + 101010)
- RX = 11 × (E0E0E0E1 + 10101010)
- R10 = 11 × (E0E0E0E0E1 + 1010101010)
Example of numbers containing only 0 and 1
n | (10n/2 − 1) / E | 10n/2 + 1 | |
---|---|---|---|
R2 | 1 | 1 × 11 | 11 |
R4 | 11 | 11 × 101 | 5 × 25 |
R6 | 111 | 111 × 1001 | 7 × 11 × 17 |
R8 | 5 × 11 × 25 | 1111 × 10001 | 75 × 175 |
RX | 11111 | 11111 × 100001 | 11 × E0E1 |
R10 | 7 × 11 × 17 × 111 | 111111 × 1000001 | 5 × 25 × EE01 |
R12 | 46E × 2X3E | 1111111 × 10000001 | 11 × 157 × 7687 |
R14 | 5 × 11 × 25 × 75 × 175 | 11111111 × 100000001 | 15 × 81 × 106X95 |
R16 | 31 × 111 × 3X891 | 111111111 × 1000000001 | 7 × 11 × 17 × E61 × 1061 |
R18 | 11 × E0E1 × 11111 | 1111111111 × 10000000001 | 52 × 25 × 24727225 |
R1X | E × 1E × 754E2E41 | 11111111111 × 100000000001 | 11 × E0E0E0E0E1 |
R20 | 5 × 7 × 11 × 17 × 25 × 111 × EE01 | 111111111111 × 1000000000001 | 75 × 141 × 175 × 8E5281 |
n | |
---|---|
R1 | 1 × 1 |
R2 | 1 × 11 |
R3 | 1 × 111 |
R4 | 1 × 1111 |
11 × 101 | |
R5 | 1 × 11111 |
R6 | 1 × 111111 |
11 × 10101 | |
111 × 1001 | |
R7 | 1 × 1111111 |
R8 | 1 × 11111111 |
11 × 1010101 | |
1111 × 10001 | |
R9 | 1 × 111111111 |
111 × 1001001 | |
RX | 1 × 1111111111 |
11 × 101010101 | |
11111 × 100001 | |
RE | 1 × 11111111111 |
R10 | 1 × 111111111111 |
11 × 10101010101 | |
111 × 1001001001 | |
1111 × 100010001 | |
111111 × 1000001 |
n | ||
---|---|---|
R6 | 1 × 111 × 1001 | E1 × 11 |
R10 | 11 × 10101 × 1000001 | EE01 × 101 |
R16 | 111 × 1001001 × 1000000001 | EEE001 × 1001 |
R20 | 1111 × 100010001 × 1000000000001 | EEEE0001 × 10001 |
n | ||
---|---|---|
R4 | 11 × 101 | |
R8 | 101 × 110011 | |
R10 | 1001 × 111000111 | 1221001221 × E1 |
R14 | 10001 × 111100001111 | |
R18 | 100001 × 111110000011111 | 1222210000122221 × E0E1 |
R20 | 1000001 × 111111000000111111 | 1221001221001221001221 × E1 |
Factorization of dozenal repunits
(Prime factors colored red means "new prime factors", i.e. the prime factors dividing Rn but not dividing Rk for any k < n)
n | factorization of Rn | number of prime factors of Rn (counted with multiplicity) | number of distinct prime factors of Rn | number of "new prime factors" (i.e. the prime factors dividing Rn but not dividing Rk for any k < n) of Rn |
1 | 1 | 0 | 0 | 0 |
2 | 11 | 1 | 1 | 1 |
3 | 111 | 1 | 1 | 1 |
4 | 5 × 11 × 25 | 3 | 3 | 2 |
5 | 11111 | 1 | 1 | 1 |
6 | 7 × 11 × 17 × 111 | 4 | 4 | 2 |
7 | 46E × 2X3E | 2 | 2 | 2 |
8 | 5 × 11 × 25 × 75 × 175 | 5 | 5 | 2 |
9 | 31 × 111 × 3X891 | 3 | 3 | 2 |
X | 11 × E0E1 × 11111 | 3 | 3 | 1 |
E | E × 1E × 754E2E41 | 3 | 3 | 3 |
10 | 5 × 7 × 11 × 17 × 25 × 111 × EE01 | 7 | 7 | 1 |
11 | 1E0411 × 69X3901 | 2 | 2 | 2 |
12 | 11 × 157 × 46E × 2X3E × 7687 | 5 | 5 | 2 |
13 | 51 × 111 × 471 × 57E1 × 11111 | 5 | 5 | 3 |
14 | 5 × 11 × 15 × 25 × 75 × 81 × 175 × 106X95 | 8 | 8 | 3 |
15 | X9X9XE × 126180EE0EE | 2 | 2 | 2 |
16 | 7 × 11 × 17 × 31 × 111 × E61 × 1061 × 3X891 | 8 | 8 | 2 |
17 | 1111111111111111111 | 1 | 1 | 1 |
18 | 52 × 11 × 25 × E0E1 × 11111 × 24727225 | 7 | 6 | 1 |
19 | 111 × 46E × 2X3E × E00E00EE0EE1 | 4 | 4 | 1 |
1X | E × 11 × 1E × 754E2E41 × E0E0E0E0E1 | 5 | 5 | 1 |
1E | 3E × 78935EX441 × 523074X3XXE | 3 | 3 | 3 |
20 | 5 × 7 × 11 × 17 × 25 × 75 × 111 × 141 × 175 × EE01 × 8E5281 | E | E | 2 |
21 | 11111 × 1277EE × 9X06176590543EE | 3 | 3 | 2 |
22 | 112 × 67 × 18X31 × X8837 × 1E0411 × 69X3901 | 7 | 6 | 3 |
23 | 31 × 111 × 3X891 × 129691 × 9894576430231 | 5 | 5 | 2 |
24 | 5 × 11 × 25 × 157 × 46E × 481 × 2X3E × 7687 × 2672288X41 | 9 | 9 | 2 |
25 | 4E × 123EE × 15960E × 160605E10497012E4E | 4 | 4 | 4 |
26 | 7 × 11 × 17 × 27 × 51 × 111 × 2E1 × 471 × 57E1 × E0E1 × 11111 × 18787 | 10 | 10 | 3 |
27 | 271 × 365E0031 × 464069563E × 39478E3664E | 4 | 4 | 4 |
28 | 5 × 11 × 15 × 25 × 75 × 81 × 175 × 75115 × 106X95 × 1748E3674115 | X | X | 2 |
29 | E × 1E × 111 × 368E51 × 2013881 × 754E2E41 × 16555E1X1 | 7 | 7 | 3 |
2X | 11 × 1587 × X9X9XE × 126180EE0EE × 7605857409257 | 5 | 5 | 2 |
2E | 5E × 34E × 46E × 2X3E × 11111 × 32XXE1 × 205812E × EX59849E | 8 | 8 | 5 |
30 | 5 × 7 × 11 × 17 × 25 × 31 × 61 × 111 × E61 × 1061 × EE01 × 3X891 × 1E807X62E61 | 11 | 11 | 2 |
31 | 1398641 × 9E2X6732EE74552406X78E76247691 | 2 | 2 | 2 |
32 | 11 × 1XE7 × 4901 × 127543624027 × 1111111111111111111 | 5 | 5 | 3 |
33 | 111 × 19491 × 1E0411 × 5XE48X1 × 69X3901 × 1064119E745041 | 6 | 6 | 3 |
34 | 52 × 11 × 25 × 35 × 75 × 175 × 375 × E0E1 × 11111 × 62041 × 1X7X9741 × 24727225 | 11 | 10 | 4 |
35 | 6E × 472488E21 × 4E2EX47X7863X18E5E18253377315E | 3 | 3 | 3 |
36 | 72 × 11 × 17 × 37 × 111 × 157 × 46E × 2X3E × 7687 × 9X17 × 76E077 × E00E00EE0EE1 | 11 | 10 | 3 |
37 | 2EE × 4159911 × 273263674E × 4X748X0X65EXX3943375X351 | 4 | 4 | 4 |
38 | 5 × E × 11 × 1E × 25 × 1461 × 2181 × 3801 × 754E2E41 × E0E0E0E0E1 × 113006390X1 | E | E | 4 |
39 | 31 × 51 × 111 × 471 × 57E1 × 11111 × 15991 × 3X891 × 1905201 × 7229231 × 7843701 | E | E | 4 |
3X | 11 × 3E × 591 × 7231 × 78935EX441 × 523074X3XXE × 3266712021E531E1 | 7 | 7 | 3 |
3E | 832966217X8X111 × 16EE6202E02X5311278504010EX13001 | 2 | 2 | 2 |
40 | 5 × 7 × 11 × 15 × 17 × 25 × 75 × 81 × 111 × 141 × 175 × 4541 × EE01 × 1E601 × 106X95 × 8E5281 × 146609481 | 15 | 15 | 3 |
41 | 46E × 2X3E × 38E01 × 3257955345X23186304E321167X366X2593101 | 4 | 4 | 2 |
42 | 11 × 1167 × E0E1 × 11111 × 2E0X1 × 36967 × 1277EE × 102X155X1 × 9X06176590543EE | 9 | 9 | 4 |
43 | 111 × X9X9XE × 6E50611 × 126180EE0EE × 16EEX75X0381X76766164247X1 | 5 | 5 | 2 |
44 | 5 × 112 × 25 × 45 × 67 × 485 × 18X31 × X8837 × 1E0411 × 69X3901 × 6X78XX0721X7626395X1 | 10 | E | 3 |
45 | 8E × 51E × 698X51 × 9X6X571 × 41X866326E31 × 191017735473010X59437791 | 6 | 6 | 6 |
46 | 7 × 11 × 17 × 31 × 91 × 111 × 1X7 × 347 × E61 × 1061 × 1X761 × 3X891 × 129691 × 13X29831 × 9894576430231 | 13 | 13 | 5 |
47 | E × 1E × 3081 × 11111 × 754E2E41 × 417569335X9871 × X54106288E1178834252681 | 7 | 7 | 3 |
48 | 5 × 11 × 25 × 75 × 157 × 175 × 46E × 481 × 2X3E × 7687 × 2672288X41 × EEEE0000EEEE0000EEEE0001 | 10 | 10 | 1 |
49 | 111 × 55E81 × 630E1 × 2376268253771 × 180218815260491 × 1111111111111111111 | 6 | 6 | 4 |
4X | 11 × 4E × 123EE × 15960E × 331887791 × 160605E10497012E4E × 348EX480E981E13E8E21 | 7 | 7 | 2 |
4E | 1092E × 13167124X1E5E6E24E1 × 991027128X344E58009X13377728X377816E | 3 | 3 | 3 |
50 | 52 × 7 × 11 × 17 × 25 × 27 × 51 × 111 × 2E1 × 471 × 57E1 × E0E1 × EE01 × 11111 × 18787 × 24727225 × 100EEEXEXEE000101 | 16 | 15 | 1 |
51 | 6X3531 × 23E214X01 × 9X134140XE002419X65090184E86425024X210X79953X1 | 3 | 3 | 3 |
52 | 11 × 271 × 50E237 × 365E0031 × 464069563E × 39478E3664E × 2221710X303X15671780413X7 | 7 | 7 | 2 |
53 | 31 × 111 × 46E × 2X3E × 3X891 × E00E00EE0EE1 × EEE000000EEE000000EEEEEE000EEEEEE001 | 7 | 7 | 1 |
54 | 5 × 11 × 15 × 25 × 75 × 81 × 175 × 541 × 75115 × 106X95 × 1748E3674115 × 22E6E3E4614X2X0E739X6493730X81 | 10 | 10 | 2 |
55 | XE × 11111 × 1E0411 × 69X3901 × 45E152651 × 328X222960EE4296E × X003996X2X7736694E501 | 7 | 7 | 4 |
56 | 7 × E × 11 × 17 × 1E × 57 × 111 × 147 × X11 × 3207 × 368E51 × 2013881 × 754E2E41 × 16555E1X1 × 76E4545077 × E0E0E0E0E1 | 14 | 14 | 5 |
57 | 13286641 × 3176XE592E × 118357X16417E44E × 2X7304X3E254X17927452584836E1769951 | 4 | 4 | 4 |
58 | 5 × 11 × 25 × 1587 × X9X9XE × 126180EE0EE × 1349X9E47X1 × 7605857409257 × 93503726E44887X0575721 | 9 | 9 | 2 |
59 | 3E × 111 × 78935EX441 × 523074X3XXE × 29E724E34313E08941 × 3X7E21516142789517140EE6X71 | 6 | 6 | 2 |
5X | 11 × 5E × 157 × 34E × 46E × 2X3E × 7687 × E0E1 × 11111 × 32XXE1 × 205812E × EX59849E × 10EEEXXXE011110EXXXE00011 | 11 | 11 | 1 |
5E | 2E61 × E411 × 48214E17105E9X31322076E52260904296018248289388E3E03E1E49225X961 | 3 | 3 | 3 |
60 | 5 × 7 × 11 × 17 × 25 × 31 × 61 × 75 × 111 × 141 × 175 × E61 × 1061 × EE01 × 3X891 × 8E5281 × 8E7701 × 86X069E01 × 1X59306601 × 1E807X62E61 | 18 | 18 | 3 |
61 | 33553026E085851E × 3E944878XE2950EX202931666X2779X976062X268389208736281XE8E | 2 | 2 | 2 |
62 | 11 × 1398641 × 19122X7 × 5705544E4727 × 11648541252X9133271 × 9E2X6732EE74552406X78E76247691 | 6 | 6 | 3 |
63 | 51 × 111 × 421 × 471 × 57E1 × 11111 × 1277EE × 22XX6541 × 9X06176590543EE × 13479X713133E72X42X33465X235661 | X | X | 3 |
64 | 5 × 11 × 25 × 1XE7 × 4901 × 7E45X1 × 127543624027 × 17962521E22801 × XXXX48E48508X7121 × 1111111111111111111 | X | X | 3 |
65 | E × 1E × 46E × 2811 × 2X3E × 754E2E41 × X6E63531 × 3108049E09E × 33473888220964X1 × 5635793E280446201E1084E | X | X | 5 |
66 | 7 × 112 × 17 × 67 × 111 × 221 × 5E7 × 18X31 × 19491 × X8837 × 1E0411 × 5XE48X1 × 69X3901 × 3547080331 × 35X1236E57 × 1064119E745041 | 15 | 14 | 4 |
67 | 79991 × 3784511E × 562747X052E4X17641X27X25E9EX5E563E27602431946E43634793X292763X675E | 3 | 3 | 3 |
68 | 52 × 11 × 15 × 25 × 35 × 75 × 81 × 175 × 375 × E0E1 × 11111 × 62041 × 106X95 × 1X7X9741 × 24727225 × EEEEEEEE00000000EEEEEEEE00000001 | 15 | 14 | 1 |
69 | 31 × 111 × 3X891 × 129691 × 1EX4391 × 9894576430231 × 6206698749E301 × E89X7X727394X9E4477E48607435213231 | 8 | 8 | 3 |
6X | 11 × 6E × 7151 × 3E022X7X7 × 472488E21 × 1646X9EEE9941 × 31404E9E46699E07 × 4E2EX47X7863X18E5E18253377315E | 8 | 8 | 4 |
6E | 11E × 6E1 × 11X1 × 1093E × 136X04861 × 69E54141E1 × 38148576964E × 79E54E87686014E × 8EE2415483165180830X595031 | 9 | 9 | 9 |
70 | 5 × 72 × 11 × 17 × 25 × 37 × 111 × 157 × 46E × 481 × 2X3E × 7687 × 9X17 × EE01 × 76E077 × 2672288X41 × E00E00EE0EE1 × 100EEEXEE0000EEEXEE000101 | 17 | 16 | 1 |
71 | 711 × 10X7E × 11111 × X9X9XE × 17508X1 × 126180EE0EE × 3378XE61E31 × 14X23716X5359331E9241 × 238566323E73EE11537XE | 9 | 9 | 6 |
72 | 11 × 2EE × 35261 × 4159911 × X4728791 × 1EX866X721 × 273263674E × 1X5729993X7EX28E3E161 × 4X748X0X65EXX3943375X351 | 9 | 9 | 4 |
73 | 4E × 111 × 251 × 1011 × 123EE × 15960E × 12478805981 × 50970299309727371 × 160605E10497012E4E × 8XX2X32378X50722348X7721 | X | X | 5 |
74 | 5 × E × 11 × 1E × 25 × 75 × 175 × 1461 × 2181 × 3801 × 1X7X6335 × 754E2E41 × E0E0E0E0E1 × 113006390X1 × 64323X6685306022E8102E16XX12046E5 | 13 | 13 | 2 |
75 | 12E × 12X1 × 59E3E × 743494471 × 10E747484336X6E1 × 233553E3X2XX2X41 × 96E647633876838EE5E × 126E002945E107EX8205E | 8 | 8 | 8 |
76 | 7 × 11 × 17 × 27 × 31 × 51 × 111 × 131 × 2E1 × 471 × E61 × 1061 × 57E1 × E0E1 × 11111 × 15991 × 18787 × 3X891 × 1905201 × 7229231 × 7843701 × 91E0577131 × 1062061062061 | 1E | 1E | 3 |
77 | 46E × 2X3E × 1E0411 × 69X3901 × 15163345X7587E × 118897E3050E0E2X6X0486585E × 68X1921E6396X5666EEE98X1E88911X711 | 7 | 7 | 3 |
78 | 5 × 11 × 25 × 3E × 591 × 7231 × 78935EX441 × 523074X3XXE × 3266712021E531E1 × EE00EE00EE00EE00EE00EE00EE00EE00EE00EE00EE01 | X | X | 1 |
79 | 111 × 271 × 5201 × 509961 × 365E0031 × 3178744351 × 464069563E × 39478E3664E × 17362950433E533544947E563334X8075359X55301 | 9 | 9 | 4 |
7X | 11 × X541 × 4176937 × 4235368181 × 832966217X8X111 × 89898362353285XE87E22X3EX7 × 16EE6202E02X5311278504010EX13001 | 7 | 7 | 4 |
7E | 13E × 11111 × 1111111111111111111 × 8362843326116X29E1610998123X98060533261X67E8E8EX5X9820154X075944023X8E | 4 | 4 | 2 |
80 | 5 × 7 × 11 × 15 × 17 × 25 × 75 × 81 × 111 × 141 × 175 × 4401 × 4541 × EE01 × 1E601 × 75115 × 106X95 × 8E5281 × 1067281 × 146609481 × 1748E3674115 × 2792X26182722E667681941 | 1X | 1X | 3 |
81 | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 | 1 | 1 | 1 |
82 | 11 × 157 × 46E × 2X3E × 7687 × 38E01 × 221X799347E672054XE87 × 5608XX9383413481532207 × 3257955345X23186304E321167X366X2593101 | 9 | 9 | 2 |
83 | E × 1E × 31 × 111 × 291 × 1X01 × 3X891 × 368E51 × 2013881 × 754E2E41 × 16555E1X1 × 9703244140E09362XX4E5E4391 × 2E77660739X567436X957763E3X61 | 11 | 11 | 4 |
84 | 53 × 11 × 25 × 85 × 841 × 1167 × E0E1 × 11111 × 2E0X1 × 36967 × 1277EE × 24727225 × 102X155X1 × 1768XX881 × 3185EE282985 × 9X06176590543EE × E64944E54364E05 | 17 | 15 | 5 |
85 | 13X34E44X48463XE × 9XX69515E88505036X5X8481573922103434716847930357X727X45X490EX3015091808814E0270X096EE | 2 | 2 | 2 |
86 | 7 × 11 × 17 × 87 × 111 × 217 × 1587 × X9X9XE × 6E50611 × 126180EE0EE × 7605857409257 × 367XE43X31E761 × 24915461720E701 × 16EEX75X0381X76766164247X1 | 12 | 12 | 4 |
87 | 2191 × 4361 × 30671E561 × 3780622808218059660EE × E536195X79679E64X9X7250756EEEE × 17440E0742X924600X5E1619X975XEXX14341 | 6 | 6 | 6 |
88 | 5 × 112 × 25 × 45 × 67 × 75 × 175 × 485 × 18X31 × X8837 × 11E241 × 1E0411 × 69X3901 × 85653E5 × 6X78XX0721X7626395X1 × 127X6X3650767881EE41272X49EXE3X919275 | 15 | 14 | 3 |
89 | 51 × 5E × 111 × 34E × 46E × 471 × 1561 × 2X3E × 57E1 × 11111 × 60E01 × 32XXE1 × 205812E × EX59849E × 1X2X6E6951 × E00E00EE0EE1 × 968035393909E245250919E58637721 | 15 | 15 | 4 |
8X | 11 × 8E × 51E × 1X11 × 17181 × 698X51 × 9X6X571 × 21XE0377 × 41X866326E31 × 191017735473010X59437791 × 18E73X9E7319678X467E056131195024481537 | E | E | 4 |
8E | 60E3E × 17565X300E × 13E2717XX7627287E54X06115112E0E1707454800170172854268382989656X12601X667825E0E2299X1E53EE0261 | 3 | 3 | 3 |
90 | 5 × 7 × 11 × 17 × 25 × 31 × 61 × 91 × 111 × 1X7 × 301 × 347 × E61 × 1061 × 2301 × EE01 × 1X761 × 3X891 × 129691 × 12387901 × 13X29831 × 1E807X62E61 × 27EX0644X61 × 683E6EE785X61 × 9894576430231 | 21 | 21 | 5 |
91 | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 | 1 | 1 | 1 |
92 | E × 11 × 1E × 237 × 3081 × 3767 × E0E1 × 11111 × 754E2E41 × E0E0E0E0E1 × 417569335X9871 × X78E8X09362X4X67 × 190E6445E1349XE4387 × X54106288E1178834252681 | 12 | 12 | 4 |
93 | 111 × 1398641 × 9E2X6732EE74552406X78E76247691 × E00E00E00E00E00E00E00E00E00E00E00E00EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE1 | 4 | 4 | 1 |
94 | 5 × 11 × 15 × 25 × 75 × 81 × 95 × 157 × 175 × 46E × 481 × 1295 × 2X3E × 7687 × E9755 × 106X95 × 215415 × 2672288X41 × 15396X698X896281 × 415E02X229X8104401 × EEEE0000EEEE0000EEEE0001 | 19 | 19 | 6 |
95 | 16E × X03E47E088EE × 2E550E66X7197E × 24386950X07374EX744E × 1513E54025X9EX58802913E8329698X84X3999E5X739X6119863E9E1E016486291 | 5 | 5 | 5 |
96 | 7 × 11 × 172 × 111 × 171 × 1XE7 × 4901 × 30351 × 55E81 × 630E1 × 127543624027 × 43404X297987 × 2376268253771 × 180218815260491 × 49535104455502961 × 1111111111111111111 | 15 | 14 | 4 |
97 | 3E × 156E × 11111 × 78935EX441 × 523074X3XXE × 761556614942428313X650583XX7608656751269E9X82X657EX43057655E34E624346133866E6EE04805E | 6 | 6 | 2 |
98 | 5 × 11 × 252 × 4E × 8X5 × 123EE × 15960E × 163065 × 331887791 × 109E77147E205 × 28XE986X778834661 × 160605E10497012E4E × 15E4751582927483X81 × 348EX480E981E13E8E21 | 13 | 12 | 5 |
99 | 31 × 111 × 3961 × 19491 × 3X891 × 1E0411 × 5XE48X1 × 69X3901 × 1064119E745041 × 729X1E8945E631 × 98717158913E7756714X091 × 658X50E23679083391136E114X124461 | 10 | 10 | 4 |
9X | 11 × 4E1 × 747 × 1092E × 4X82344661 × 146987541X202897 × 13167124X1E5E6E24E1 × 6600348E4E8666E8417207808371 × 991027128X344E58009X13377728X377816E | 9 | 9 | 5 |
9E | 17E × 46E × 9E1 × 2X3E × 67401 × X9X9XE × 7891X7E × 126180EE0EE × 1X6E953XX0X569X48034566217EXX38651087X821EEX4XX846X1X39151536560E1230267X513EX41 | 9 | 9 | 5 |
X0 | 52 × 7 × 11 × 17 × 25 × 27 × 35 × 51 × 75 × 111 × 141 × 175 × 181 × 2E1 × 375 × 471 × 57E1 × E0E1 × EE01 × 11111 × 18787 × 62041 × 8E5281 × 1X7X9741 × 24727225 × 100EEEXEXEE000101 × 72066X5588687E7205E79986211241 | 24 | 23 | 2 |
X1 | E2 × 1E × 408E × 754E2E41 × 1E2347X16E × 273389373E × 26068098770838687570277297428517E3623E521 × 3097108E5XXEE91692504067248816X526X43908150E871 | 9 | 8 | 5 |
X2 | 11 × 54991 × 6X3531 × 23E214X01 × 151540EX01 × 6E9704475971 × 25785880331329317729818949936575071 × 9X134140XE002419X65090184E86425024X210X79953X1 | 8 | 8 | 4 |
X3 | 6E × 111 × 472488E21 × 4E2EX47X7863X18E5E18253377315E × E00E00E00E00E00E00E00E00E00E00E00E00E00E0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE1 | 5 | 5 | 1 |
X4 | 5 × 11 × 25 × 271 × 259441 × 50E237 × 365E0031 × 464069563E × 39478E3664E × 3012X78X7X0X1 × 3373558414615602861 × 5977650E39345191054X8941 × 2221710X303X15671780413X7 | 11 | 11 | 4 |
X5 | 18E × 7E5E × 11111 × 5887E × 1277EE × 9X06176590543EE × 19906116507647451X176848466145277E42E8274EX793924127713152370055529E473202388800E17E01370E | 7 | 7 | 4 |
X6 | 72 × 11 × 17 × 31 × 37 × X7 × 111 × 157 × 46E × E61 × 1061 × 2X3E × 7687 × 9X17 × 3X891 × 76E077 × 22671E17 × 5X9815491 × E00E00EE0EE1 × 1062060EEE001062061 × EEE000000EEE000000EEEEEE000EEEEEE001 | 1X | 19 | 4 |
X7 | 1524X46E41E × XE40396414558012072859E47584E1E × X01E6709327152983887984680242642205X31132061454580217E479100E22X553675016298052383851 | 3 | 3 | 3 |
X8 | 5 × 11 × 15 × 25 × 75 × 81 × 175 × 541 × 18X81 × 3E055 × 75115 × 106X95 × 1E395315 × 1748E3674115 × 491E191565059401 × 22E6E3E4614X2X0E739X6493730X81 × 2344208779XX6258285840X8052004X81 | 15 | 15 | 5 |
X9 | 111 × 2EE × 1326391 × 4159911 × 273263674E × 4X748X0X65EXX3943375X351 × 8824771150225312417191041661931977572125372E14654697446827187452864604E8535221 | 7 | 7 | 2 |
XX | 112 × 67 × XE × E0E1 × 11111 × 18X31 × 19X87 × X8837 × 1E0411 × 69X3901 × 158X0217 × 45E152651 × 328X222960EE4296E × X003996X2X7736694E501 × 49X2886048599849242952388618X39019461 | 14 | 13 | 3 |
XE | 19E × 72018384965970451E067536910118X32E1123514585X8208319E4E223E82731346976E3357E3X060X5525748EXE5590563737X2842E7804E892129EE603X70E | 2 | 2 | 2 |
E0 | 5 × 7 × E × 11 × 17 × 1E × 25 × 57 × 111 × 147 × X11 × 1461 × 2181 × 3207 × 3801 × EE01 × 53301 × X7141 × 368E51 × 6X99X1 × 2013881 × 754E2E41 × 16555E1X1 × 76E4545077 × E0E0E0E0E1 × 113006390X1 × 4622E8098X921X0916X79XXX1 | 23 | 23 | 4 |
E1 | 46E × 2X3E × 32961 × X2645EEE × 1111111111111111111 × 1E199E5X1154899267903XX31 × 20X584263846XE267E0734E2X71X15426X58EEX131091169959376E267X50562XE0E379E | 7 | 7 | 4 |
E2 | 11 × 219X7 × 926337 × 13286641 × 3176XE592E × 118357X16417E44E × 2X7304X3E254X17927452584836E1769951 × 68589X43X7633909564843X29E049E4413X711E90945851378E91E01 | 8 | 8 | 3 |
E3 | 31 × 51 × 111 × 471 × 57E1 × 11111 × 15991 × 3X891 × 129691 × 1905201 × 7229231 × 7843701 × 9894576430231 × 3082666E0496E4611X161 × EE16650679438E4210231 × 3E484218E44X9828444735035540E31 | 14 | 14 | 3 |
E4 | 5 × 11 × 25 × 75 × E5 × 175 × 1587 × X9X9XE × 336E8XX001 × 126180EE0EE × 1349X9E47X1 × 7605857409257 × 93503726E44887X0575721 × 39X756215584887E1214176X2007106E6730624521EX071948935 | 12 | 12 | 3 |
E5 | 278888E × 6559E89566E16861 × 35X0822EX99E414205806454EE0917851692858E × 27805997X962266E9XX8X3X504E199E7491540342X14E313791X72X00627776026XX68E1X11 | 4 | 4 | 4 |
E6 | 7 × 11 × 17 × 3E × E7 × 111 × 1E1 × 591 × 4X57 × 7231 × 7671581 × 374594E7 × 78935EX441 × 523074X3XXE × 3266712021E531E1 × 29E724E34313E08941 × 76E45076E4545076E45077 × 3X7E21516142789517140EE6X71 | 16 | 16 | 6 |
E7 | 6442E324EE × 5759X3929EE8E112931 × 39X37X50E66476X58688XX15390622XX480X471396596E × 11950E73490699E9X211171624E426E56905E3846326383X686767093X2281E51 | 4 | 4 | 4 |
E8 | 52 × 11 × 25 × 5E × 157 × 34E × 46E × 481 × 2X3E × 7687 × E0E1 × 11111 × 32XXE1 × 205812E × 200X2121 × 24727225 × EX59849E × 201347X741 × 2672288X41 × 130982376E76481 × 247X950907970707X1 × 10EEEXXXE011110EXXXE00011 | 1E | 1X | 4 |
E9 | 111 × 120EX31 × 400177X625E2E91 × 832966217X8X111 × 16EE6202E02X5311278504010EX13001 × 24168717605137390789685298X0709447047E8957264038813319910978876E242X5XE1 | 6 | 6 | 3 |
EX | 11 × 2E61 × E411 × 54820E8449361 × 207E168E81X24231305XEX70916E7325811X6394X492811XE4311E2351 × 48214E17105E9X31322076E52260904296018248289388E3E03E1E49225X961 | 6 | 6 | 2 |
EE | E × 1E × 1E0411 × 69X3901 × 754E2E41 × 259EE7EX1 × 42EXEEE90E70755X2317X981 × 105E2X559298E37XX0X7E6E19729857X29748931432081246479069E8512X716636446E7E2X1062948E569051 | 8 | 8 | 3 |
100 | 5 × 7 × 11 × 15 × 17 × 25 × 31 × 61 × 75 × 81 × 111 × 141 × 175 × 401 × E61 × 1061 × 4541 × EE01 × 1E601 × 3X891 × 106X95 × 8E5281 × 8E7701 × E09201 × 146609481 × 86X069E01 × 1X59306601 × 1E807X62E61 × 10257266E3854X01 × 323X555614903716989X19801 | 26 | 26 | 4 |
The product of "new prime factors" (i.e. the prime factors dividing Rn but not dividing Rk for any k < n) of Rn is Φn(10)/GCD(Φn(10),n), except n = 1 and n = E (in which cases, the two numbers are 1 and E for n = 1, and they are 11111111111 (= RE) and 123456789E for n = E)).
In fact, the repunit Rn = Πd|n, d>1(Φd(10)), where Φd(10) is the dth cyclotomic polynomial evaluated at 10.
Properties
- Any positive multiple of the repunit Rn contains at least n nonzero digits.
- The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 27 (111 in base 5, 11111 in base 2) and 48X7 (111 in base 76, 1111111111111 in base 2). The Goormaghtigh conjecture says there are only these two cases.
- Using the pigeon-hole principle it can be easily shown that for each n and b such that n and b are relatively prime there exists a repunit in base b that is a multiple of n. To see this consider repunits R1(b),...,Rn(b). Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1≤i<j≤n such that Ri(b) and Rj(b) have the same residue modulo n. It follows that Rj(b) - Ri(b) has residue 0 modulo n, i.e. is divisible by n. Rj(b) - Ri(b) consists of j - i ones followed by i zeroes. Thus, Rj(b) - Ri(b) = Rj-i(b) x bi . Since n divides the left-hand side it also divides the right-hand side and since n and b are relatively prime n must divide Rj-i(b). e.g. for every 5-rough number n, there exists a repunit in base 10 that is a multiple of n.
- The Feit–Thompson conjecture is that Rq(p) never divides Rp(q) for two distinct primes p and q.
- Using the Euclidean Algorithm for repunits definition: R1(b) = 1; Rn(b) = Rn-1(b) x b + 1, any consecutive repunits Rn-1(b) and Rn(b) are relatively prime in any base b for any n.
- If m and n are relatively prime, Rm(b) and Rn(b) are relatively prime in any base b for any m and n. The Euclidean Algorithm is based on gcd(m, n) = gcd(m - n, n) for m > n. Similarly, using Rm(b) - Rn(b) × bm-n = Rm-n(b), it can be easily shown that gcd(Rm(b), Rn(b)) = gcd(Rm-n(b), Rn(b)) for m > n. Therefore if gcd(m, n) = 1, then gcd(Rm(b), Rn(b)) = R1(b) = 1.
- If r is a divisor of b−1, then the remainder of Rn(b) modulo r is equal to the remainder of n modulo r, e.g. the remainder of Rn(10) modulo E is equal to the remainder of n modulo E.
- Repunits in base 10 are related the cyclic patterns of repeating dozenals, it was found very early on that for any prime p greater than 3 except E, the period of the dozenal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p.
- The only one repunit prime in base 4 is 5 (=114).
- The only one repunit prime in base 8 is 61 (=1118).
- The only one repunit prime in base 14 is 15 (=1114).
- The only one repunit prime in base 23 is 531 (=11123).
- The only one repunit prime in base 30 is 31 (=1130).
- The only one repunit prime in base 84 is 85 (=1184).
- The only one repunit prime in base X8 is 5E70EX5X8801 (=1111111X8).
- There are no repunit primes in bases 9, 21, 28, 41, 54, 69, X1, X5, 100, ..., because of algebraic factors, especially, all repunits in base 9 are triangular numbers (since all triangular numbers×n+1 are centered n-gonal numbers, and all centered nonagonal (9-gonal) numbers are also triangular numbers), and no triangular numbers >3 are primes.
- Every positive perfect power base has at most one generalized repunit prime (since generalized repunits in these bases can be factored algebraically), and it is conjectured that every positive non-perfect power base has infinitely many generalized repunit primes (there is at least one known repunit prime or repunit PRP with length < 3000 for all non-perfect power bases 2<=b<=100, and except base 43 (=3*15) and base 77 (=7*11), all other non-perfect power bases 2<=b<=100 have at least one known repunit prime with length < 1000, besides, base 43 and base 77 also have one known repunit prime with length < 3000, (43^2545−1)/42 and (77^2685−1)/76.
- Every negative base which is either perfect power or of the form −4k4 has at most one generalized repunit prime (since generalized repunits in these bases can be factored algebraically) (except −4, which has two generalized repunit primes: R2(−4) = −3 and R3(−4) = 11), and it is conjectured that every negative base which is neither perfect power nor of the form −4k4 has infinitely many generalized repunit primes.
Bases such that the repunit numbers have algebraic factors
Positive bases
- all such bases (exactly the perfect powers): 4, 8, 9, 14, 21, 23, 28, 30, 41, 54, 69, 84, X1, X5, X8, 100, 121, 144, 160, 169, 183, 194, 201, 230, 247, 261, 294, 309, 344, 368, 381, 400, 441, 484, 509, 554, 5X1, 630, 681, 6E4, 714, 769, 804, 861, 900, 92E, 961, X04, X69, E14, E81, 1000, 1030, 10X1, 1154, 1209, 1228, 1284, 1323, 1331, 1341, 1400, 1481, 1544, 1609, 1694, 1708, 1761, 1830, 1901, 1985, 1994, 1X69, 1E44, 1E53, 2021, 2100, 21X1, 2284, 2369, 2454, 2541, 2630, 2721, 2814, 2909, 2X04, 2X15, 2E01, 3000, 3101, 3204, 3309, 3414, 3460, 3521, 3630, 3741, 3854, 3969, 3X84, 3E77, 3EX1, 4100, 4221, 4344, 4469, 4594, 4600, 4701, 4768, 4830, 48X8, 4961, 4X94, 5009, 5144, 5281, 5400, 5439, 5541, 5684, 5809, 5954, 5XX1, 6030, 6181, 61E4, 6314, 6469, 6604, 6761, 6900, 6X61, 7004, 705E, 7169, 7314, 7481, 7630, 77X1, 7954, 7E09, 8000, 8084, 8241, 8400, 8581, 8744, 8909, 8X94, 9061, 9230, 9401, 9594, 9769, 9887, 9944, 9E21, X100, X208, X2X1, X484, X669, X854, XX41, E030, E221, E414, E483, E609, E804, EX01, 10000, ...
- all such bases with only one repunit prime (pr-th power bases with p prime and r≥1 such that Rp is prime and the only repunit prime): 4, 8, 14, 23, 30, 84, X8, 144, 194, 294, 368, 400, 484, 900, 92E, E14, 1830, 1994, 2630, 3204, 4100, 4768, 4830, 5144, 5439, 7004, 7954, 8400, 8X94, 9230, 9944, X208, X484, ...
- all such bases with no repunit primes (pr-th power bases with p prime and r≥1 such that Rp is composite, or c-th power bases with c non-primepower): 9, 21, 28, 41, 54, 69, X1, X5, 100, 121, 160, 169, 183, 201, 230, 247, 261, 309, 344, 381, 441, 509, 554, 5X1, 630, 681, 6E4, 714, 769, 804, 861, 961, X04, X69, E81, 1000, 1030, 10X1, 1154, 1209, 1228, 1284, 1323, 1331, 1341, 1400, 1481, 1544, 1609, 1694, 1708, 1761, 1901, 1985, 1X69, 1E44, 1E53, 2021, 2100, 21X1, 2284, 2369, 2454, 2541, 2721, 2814, 2909, 2X04, 2X15, 2E01, 3000, 3101, 3309, 3414, 3460, 3521, 3630, 3741, 3854, 3969, 3X84, 3E77, 3EX1, 4221, 4344, 4469, 4594, 4600, 4701, 48X8, 4961, 4X94, 5009, 5281, 5400, 5541, 5684, 5809, 5954, 5XX1, 6030, 6181, 61E4, 6314, 6469, 6604, 6761, 6900, 6X61, 705E, 7169, 7314, 7481, 7630, 77X1, 7E09, 8000, 8084, 8241, 8581, 8744, 8909, 9061, 9401, 9594, 9769, 9887, 9E21, X100, X2X1, X669, X854, XX41, E030, E221, E414, E483, E609, E804, EX01, 10000, ...
- squares: 4, 9, 14, 21, 30, 41, 54, 69, 84, X1, 100, 121, 144, 169, 194, 201, 230, 261, 294, 309, 344, 381, 400, 441, 484, 509, 554, 5X1, 630, 681, 714, 769, 804, 861, 900, 961, X04, X69, E14, E81, 1030, 10X1, 1154, 1209, 1284, 1341, 1400, 1481, 1544, 1609, 1694, 1761, 1830, 1901, 1994, 1X69, 1E44, 2021, 2100, 21X1, 2284, 2369, 2454, 2541, 2630, 2721, 2814, 2909, 2X04, 2E01, 3000, 3101, 3204, 3309, 3414, 3521, 3630, 3741, 3854, 3969, 3X84, 3EX1, 4100, 4221, 4344, 4469, 4594, 4701, 4830, 4961, 4X94, 5009, 5144, 5281, 5400, 5541, 5684, 5809, 5954, 5XX1, 6030, 6181, 6314, 6469, 6604, 6761, 6900, 6X61, 7004, 7169, 7314, 7481, 7630, 77X1, 7954, 7E09, 8084, 8241, 8400, 8581, 8744, 8909, 8X94, 9061, 9230, 9401, 9594, 9769, 9944, 9E21, X100, X2X1, X484, X669, X854, XX41, E030, E221, E414, E609, E804, EX01, 10000, ...
- square bases such that R2 is prime and the only repunit prime: 4, 14, 30, 84, 144, 194, 294, 400, 484, 900, E14, 1830, 1994, 2630, 3204, 4100, 4830, 5144, 7004, 7954, 8400, 8X94, 9230, 9944, X484, ...
- square bases with no repunit primes: 9, 21, 41, 54, 69, X1, 100, 121, 169, 201, 230, 261, 309, 344, 381, 441, 509, 554, 5X1, 630, 681, 714, 769, 804, 861, 961, X04, X69, E81, 1030, 10X1, 1154, 1209, 1284, 1341, 1400, 1481, 1544, 1609, 1694, 1761, 1901, 1X69, 1E44, 2021, 2100, 21X1, 2284, 2369, 2454, 2541, 2721, 2814, 2909, 2X04, 2E01, 3000, 3101, 3309, 3414, 3521, 3630, 3741, 3854, 3969, 3X84, 3EX1, 4221, 4344, 4469, 4594, 4701, 4961, 4X94, 5009, 5281, 5400, 5541, 5684, 5809, 5954, 5XX1, 6030, 6181, 6314, 6469, 6604, 6761, 6900, 6X61, 7169, 7314, 7481, 7630, 77X1, 7E09, 8084, 8241, 8581, 8744, 8909, 9061, 9401, 9594, 9769, 9E21, X100, X2X1, X669, X854, XX41, E030, E221, E414, E609, E804, EX01, 10000, ...
- cubes: 8, 23, 54, X5, 160, 247, 368, 509, 6E4, 92E, 1000, 1331, 1708, 1E53, 2454, 2X15, 3460, 3E77, 4768, 5439, 61E4, 705E, 8000, 9061, X208, E483, 10854, 12145, 13760, 152X7, 16E68, 18969, 1X8E4, 2098E, 23000, 25391, 27908, 2X3E3, 31054, 33X75, 36X60, 3X017, 41368, 44899, 483E4, 500EE, 54000, 58101, 60408, 64923, 69454, 721X5, 77160, 80347, 85768, 8E209, 94XE4, 9XX2E, X5000, XE431, E5E08, 100853, 107854, 112E15, 11X460, 126077, 131E68, 13X139, 1465E4, 15315E, 160000, 169161, 176608, 184183, 192054, 1X0245, 1XX760, 1E93X7, 208368, 217669, 2270E4, 236X8E, 247000, 257491, 268108, 2790E3, 28X454, 29EE75, 2E1X60, 304117, 316768, 329599, 3407E4, 3541EE, 368000, 380201, 394808, 3X9623, 402854, 4182X5, 432160, 448447, 462E68, 479E09, 4952E4, 4E0E2E, 509000, 525531, 542308, 55E553, 579054, 597015, 5E5460, 614177, 633368, 652X39, 6729E4, 69325E, 6E4000, 715261, 736X08, 758X83, 77E454, 7X2345, 805760, 8294X7, 851768, 876369, 89E4E4, 904E8E, 92E000, 955591, 980508, 9X79E3, X13854, X40075, X68X60, X96217, E03E68, E32299, E60EE4, E902EE, 1000000, ...
- cube bases such that R3 is prime and the only repunit prime: 8, 23, 368, 92E, 4768, 5439, X208, 13760, 60408, 64923, 85768, X5000, 1XX760, 394808, 432160, 652X39, 876369, 980508, ...
- cube bases with no repunit primes: 54, X5, 160, 247, 509, 6E4, 1000, 1331, 1708, 1E53, 2454, 2X15, 3460, 3E77, 61E4, 705E, 8000, 9061, E483, 10854, 12145, 152X7, 16E68, 18969, 1X8E4, 2098E, 23000, 25391, 27908, 2X3E3, 31054, 33X75, 36X60, 3X017, 41368, 44899, 483E4, 500EE, 54000, 58101, 69454, 721X5, 77160, 80347, 8E209, 94XE4, 9XX2E, XE431, E5E08, 100853, 107854, 112E15, 11X460, 126077, 131E68, 13X139, 1465E4, 15315E, 160000, 169161, 176608, 184183, 192054, 1X0245, 1E93X7, 208368, 217669, 2270E4, 236X8E, 247000, 257491, 268108, 2790E3, 28X454, 29EE75, 2E1X60, 304117, 316768, 329599, 3407E4, 3541EE, 368000, 380201, 3X9623, 402854, 4182X5, 448447, 462E68, 479E09, 4952E4, 4E0E2E, 509000, 525531, 542308, 55E553, 579054, 597015, 5E5460, 614177, 633368, 6729E4, 69325E, 6E4000, 715261, 736X08, 758X83, 77E454, 7X2345, 805760, 8294X7, 851768, 89E4E4, 904E8E, 92E000, 955591, 9X79E3, X13854, X40075, X68X60, X96217, E03E68, E32299, E60EE4, E902EE, 1000000, ...
- there are no positive 5-th power bases up to 106 such that R5 is prime, the smallest positive 5-th power base such that R5 is prime is 1886514 (=1X5)
- there are two positive 7-th power bases up to 106 such that R7 is prime: X8 (=27) and 39265 (=57)
- for any prime p>7, there are no positive p-th power bases up to 106 such that Rp is prime.
Negative bases
- all such bases (exactly the perfect odd powers plus the numbers of the form 4k4): 4, 8, 23, 28, 54, X5, X8, 160, 183, 230, 247, 368, 509, 6E4, 714, 92E, 1000, 1228, 1323, 1331, 1544, 1708, 1985, 1E53, 2454, 2X15, 3000, 3460, 3E77, 4600, 4768, 48X8, 5439, 5684, 61E4, 705E, 8000, 9061, 9594, 9887, X208, E483, ...
- all such bases with only two repunit prime (R2 and Rp with odd prime p): 4, X8, ...
- all such bases with only one repunit prime and it is R2: 8, 28, 48X8, ...
- all such bases with only one repunit prime and it is Rp with odd prime p: 160, 247, 509, 4600, 9887, ...
- all such bases with no repunit primes: 23, 54, X5, 183, 230, 368, 6E4, 714, 92E, 1000, 1228, 1323, 1331, 1544, 1708, 1985, 1E53, 2454, 2X15, 3000, 3460, 3E77, 4768, 5439, 5684, 61E4, 705E, 8000, 9061, 9594, X208, E483, ...
- odd powers: 8, 23, 28, 54, X5, X8, 160, 183, 247, 368, 509, 6E4, 714, 92E, 1000, 1228, 1323, 1331, 1708, 1985, 1E53, 2454, 2X15, 3460, 3E77, 4600, 4768, 48X8, 5439, 61E4, 705E, 8000, 9061, 9594, 9887, X208, E483, ...
- odd power bases with only two repunit prime (R2 and Rp with odd prime p): X8, ...
- odd power bases with only one repunit prime and it is R2: 8, 28, 48X8, ...
- odd power bases with only one repunit prime and it is Rp with odd prime p: 160, 247, 509, 4600, 9887, ...
- odd power bases with no repunit primes: 23, 54, X5, 183, 368, 6E4, 714, 92E, 1000, 1228, 1323, 1331, 1708, 1985, 1E53, 2454, 2X15, 3460, 3E77, 4768, 5439, 61E4, 705E, 8000, 9061, 9594, X208, E483, ...
- cubes: 8, 23, 54, X5, 160, 247, 368, 509, 6E4, 92E, 1000, 1331, 1708, 1E53, 2454, 2X15, 3460, 3E77, 4768, 5439, 61E4, 705E, 8000, 9061, X208, E483, 10854, 12145, 13760, 152X7, 16E68, 18969, 1X8E4, 2098E, 23000, 25391, 27908, 2X3E3, 31054, 33X75, 36X60, 3X017, 41368, 44899, 483E4, 500EE, 54000, 58101, 60408, 64923, 69454, 721X5, 77160, 80347, 85768, 8E209, 94XE4, 9XX2E, X5000, XE431, E5E08, 100853, 107854, 112E15, 11X460, 126077, 131E68, 13X139, 1465E4, 15315E, 160000, 169161, 176608, 184183, 192054, 1X0245, 1XX760, 1E93X7, 208368, 217669, 2270E4, 236X8E, 247000, 257491, 268108, 2790E3, 28X454, 29EE75, 2E1X60, 304117, 316768, 329599, 3407E4, 3541EE, 368000, 380201, 394808, 3X9623, 402854, 4182X5, 432160, 448447, 462E68, 479E09, 4952E4, 4E0E2E, 509000, 525531, 542308, 55E553, 579054, 597015, 5E5460, 614177, 633368, 652X39, 6729E4, 69325E, 6E4000, 715261, 736X08, 758X83, 77E454, 7X2345, 805760, 8294X7, 851768, 876369, 89E4E4, 904E8E, 92E000, 955591, 980508, 9X79E3, X13854, X40075, X68X60, X96217, E03E68, E32299, E60EE4, E902EE, 1000000, ...
- cube bases such that R2 is prime and the only repunit prime: 8 (no others)
- cube bases such that R3 is prime and the only repunit prime: 160, 247, 509, 36X60, 58101, 77160, 11X460, 169161, 217669, 28X454, 368000, 402854, 4952E4, 6729E4, 6E4000, 715261, 805760, ...
- cube bases with no repunit primes: 23, 54, X5, 368, 6E4, 92E, 1000, 1331, 1708, 1E53, 2454, 2X15, 3460, 3E77, 4768, 5439, 61E4, 705E, 8000, 9061, X208, E483, 10854, 12145, 13760, 152X7, 16E68, 18969, 1X8E4, 2098E, 23000, 25391, 27908, 2X3E3, 31054, 33X75, 3X017, 41368, 44899, 483E4, 500EE, 54000, 60408, 64923, 69454, 721X5, 80347, 85768, 8E209, 94XE4, 9XX2E, X5000, XE431, E5E08, 100853, 107854, 112E15, 126077, 131E68, 13X139, 1465E4, 15315E, 160000, 176608, 184183, 192054, 1X0245, 1XX760, 1E93X7, 208368, 2270E4, 236X8E, 247000, 257491, 268108, 2790E3, 29EE75, 2E1X60, 304117, 316768, 329599, 3407E4, 3541EE, 380201, 394808, 3X9623, 4182X5, 432160, 448447, 462E68, 479E09, 4E0E2E, 509000, 525531, 542308, 55E553, 579054, 597015, 5E5460, 614177, 633368, 652X39, 69325E, 736X08, 758X83, 77E454, 7X2345, 8294X7, 851768, 876369, 89E4E4, 904E8E, 92E000, 955591, 980508, 9X79E3, X13854, X40075, X68X60, X96217, E03E68, E32299, E60EE4, E902EE, 1000000, ...
- there are three negative 5-th power bases up to 106 such that R5 is prime: 4600 (=65), 9887 (=75), and 585815 (=155)
- there are only one negative 7-th power base up to 106 such that R7 is prime: X8 (=27)
- for any prime p>7, there are no negative p-th power bases up to 106 such that Rp is prime.
- numbers of the form 4k4: 4, 54, 230, 714, 1544, 3000, 5684, 9594, 13230, 1E194, 29X84, 40000, 56144, 74E14, 99230, 107854, 141404, 183000, 211804, 26X454, 316230, 392314, 45E944, 540000, 634284, 741994, 866230, 9X6994, E45284, 1103000, 12X1944, 14X3314, 1709230, 1959454, 2015804, 2300000, 2616404, 2962854, 3123230, 3519E14, 3951144, 4203000, 46E5X84, 5030194, 55E0230, 5EE8594, 6653684, 7140000, 7880544, 8457714, 9090230, 9965054, X6X1004, E483000, 10312004, 11211054, 12183230, 131XE714, 14295544, 15440000, 1666X684, 17964594, 19129230, 1X584194, 1EXE4X84, 21503000, 22EE2144, 24785E14, 26446230, 281E6854, 2X05E404, 30000000, 32060804, 34205454, 36476230, 38837314, 3E110944, 41703000, 44216284, 46X52994, 497E9230, 50695994, 536X9284, 56840000, 59E16944, 6133X314, 648E3230, 68402454, 70070804, 73X83000, 77X42404, 7EE73854, 84260230, 88708E14, 9113E144, 95940000, 9X714X84, X3687194, X8820230, E1E61594, E7494684, 101003000, 106932544, 110868714, 1169E3230, 121154054, 127715004, 132300000, 139117004, 144168054, 14E439230, 156954714, 162500544, 16X303000, 176366684, 182675594, 18E036230, 197873194, 1X4772X84, 1E1940000, 1EE1X1144, 208920E14, 216746230, 22485E854, 233070404, 241783000, 2505X2804, 25E716454, 26XE69230, 27X726314, 28X614944, 29X840000, 2XE1XE284, 2EEXX9994, 310E43230, 32233X994, 333XX4284, 345X03000, 3580X2944, 36X74E314, 381590230, 3947E1454, 3X81EX804, 400000000, ...
- bases of the form 4k4 with only two repunit primes: 4 (no others)
- bases of the form 4k4 with only one repunit prime: (none)
- bases of the form 4k4 with no repunit primes: 54, 230, 714, 1544, 3000, 5684, 9594, 13230, 1E194, 29X84, 40000, 56144, 74E14, 99230, 107854, 141404, 183000, 211804, 26X454, 316230, 392314, 45E944, 540000, 634284, 741994, 866230, 9X6994, E45284, 1103000, 12X1944, 14X3314, 1709230, 1959454, 2015804, 2300000, 2616404, 2962854, 3123230, 3519E14, 3951144, 4203000, 46E5X84, 5030194, 55E0230, 5EE8594, 6653684, 7140000, 7880544, 8457714, 9090230, 9965054, X6X1004, E483000, 10312004, 11211054, 12183230, 131XE714, 14295544, 15440000, 1666X684, 17964594, 19129230, 1X584194, 1EXE4X84, 21503000, 22EE2144, 24785E14, 26446230, 281E6854, 2X05E404, 30000000, 32060804, 34205454, 36476230, 38837314, 3E110944, 41703000, 44216284, 46X52994, 497E9230, 50695994, 536X9284, 56840000, 59E16944, 6133X314, 648E3230, 68402454, 70070804, 73X83000, 77X42404, 7EE73854, 84260230, 88708E14, 9113E144, 95940000, 9X714X84, X3687194, X8820230, E1E61594, E7494684, 101003000, 106932544, 110868714, 1169E3230, 121154054, 127715004, 132300000, 139117004, 144168054, 14E439230, 156954714, 162500544, 16X303000, 176366684, 182675594, 18E036230, 197873194, 1X4772X84, 1E1940000, 1EE1X1144, 208920E14, 216746230, 22485E854, 233070404, 241783000, 2505X2804, 25E716454, 26XE69230, 27X726314, 28X614944, 29X840000, 2XE1XE284, 2EEXX9994, 310E43230, 32233X994, 333XX4284, 345X03000, 3580X2944, 36X74E314, 381590230, 3947E1454, 3X81EX804, 400000000, ...
List of repunit (probable) primes base b
We also consider “negative primes” (e.g. R2(−10) = −E) as primes, since they are primes in the domain Z (the set of all integers).
b | length of repunit (probable) primes in base b |
---|---|
−100 | 3, 1E, 35, 225, 1E4E, 22237, 59307, ... |
−EE | 7, 15, 17, 3E, 87, 2687, X6EE, ... |
−EX | 3, 4441, ... |
−E9 | 5, X27, ... |
−E8 | 2, 4E, 1530E, ... |
−E7 | 3, 15, 3E, 1677, 16X7, 13371, ... |
−E6 | 2, 87, 401, 62X5, ... |
−E5 | 85, 181, 255, 11X7, 1078E, ... |
−E4 | 5, 7, 1E, 4E, 147, 1231, 3617, ... |
−E3 | 5, 7, 1667, 6E01, ... |
−E2 | 11, 817, 3X91, ... |
−E1 | 5, 7, 15, 4E, 67, 111, ... |
−E0 | 2, 3, 85, 111, 907, ... |
−XE | 5, 85, 1E65, 20X5, ... |
−XX | 32E, ... |
−X9 | 15, 16E, 1021, ... |
−X8 | 2, 7 (no others) |
−X7 | 225, 745, 119X7, ... |
−X6 | 5, 11, 3E, 117, 17E, 274E, ... |
−X5 | (none) |
−X4 | 960E, ... |
−X3 | 25, 517, ... |
−X2 | 205, 22E1, 7561, 10X85, ... |
−X1 | 5, 11, 81, X4E, 6675, 13556E, ... |
−X0 | 3, 27, 37, 19E, 2X1E, ... |
−9E | 25, 45, 565, 6797, ... |
−9X | 3, 1E, 91, 1445, ... |
−99 | 1X7, ... |
−98 | 95, X35, 1261, 9935, ... |
−97 | 7, 27, 205, ... |
−96 | 2, 7, 11, 1061, 7287, ... |
−95 | ... |
−94 | 3, ... |
−93 | 3, 5, 1E, 45, 27E, 121E, 7011, ... |
−92 | 2, 1E, 85, 9X41, ... |
−91 | 4E, 67, 577, ... |
−90 | 2, 11, 167, 912E, ... |
−8E | 87, 69E, X541, 1473E, ... |
−8X | 3, 7, 17, 1E, 27, 2385, ... |
−89 | E, 105, 82E, E37, ... |
−88 | 2, 481, 59E, 71E, 12491, 143X1, ... |
−87 | ... |
−86 | 2, 3, ... |
−85 | 7, 171, 44E1E, 80681, ... |
−84 | 3, 205, 325, 6X4E, 44175, ... |
−83 | 7, 31, 35, 5E, ... |
−82 | 2, 17, 85, 39725, 56577, 91897, ... |
−81 | ... |
−80 | 31, 87, XE, 19E, 16895, ... |
−7E | 37, 46055, 619EE, ... |
−7X | 5E, 217, 431, 104E, 2241, 601E, ... |
−79 | 75, 3E7, 421, 22E1, ... |
−78 | 31, 4E, 95, ... |
−77 | 3, E, 37, 291, 10561, 19857, 2E987, ... |
−76 | 2, 3, 3E, ... |
−75 | 11, 4E, E5, 77E, 2687, 3E981, 4X7XE, ... |
−74 | 4E1, 965, 2E89E, X096E, ... |
−73 | 7, 32E, 20EE1, ... |
−72 | 7, 15, 291, 4187, 4EX67, 5E6X5, ... |
−71 | 11E, 2065, 24205, 6829E, ... |
−70 | 2, 7, 11, E7, 25E, 68E, 767, 205E, ... |
−6E | 17, 27, 31, 37, 2E1, 397, 1911, 5147, ... |
−6X | 205, 8X7, 4827E, ... |
−69 | 3, 5, 4X5, 591, 71E, 721, 4225, E31E, ... |
−68 | 2, 5, 11, 16E, 307, 93101, 93231, E022E, ... |
−67 | 3, 8E, 321, 34E, 11E7, ... |
−66 | 3, 7, 27, 471, 2535, ... |
−65 | 31, 225, ... |
−64 | 3, 5, 13E, 1X5, 11771, 8004E, ... |
−63 | 5, 6E, 3717, ... |
−62 | 2, 11, 27, 31, 91, X087, 809X7, ... |
−61 | 7, 1X811, ... |
−60 | 2, 3, 7, 67, 1E1, 197E, ... |
−5E | 5, 31, 311E, 440E, 337E7, 39001, ... |
−5X | 3, 51, 81, 7907, 20895, ... |
−59 | E, 157, 17E, 285, 35E, 2835, 12467, ... |
−58 | 2, 531, 545, 35601, ... |
−57 | 3, 1437, 1825, 1X2E, ... |
−56 | 7, 15, 157, 457, 148X1, 29EE1, 30607, 32E85, ... |
−55 | 17, 27, ... |
−54 | (none) |
−53 | 3, 31, 35, 1297, 23E7, 10X8E, 25927, 4E107, 90X6E, E6015, ... |
−52 | 2, E, 25, 11E, 221, 9707, 1X48E, 82837, ... |
−51 | 7, 35, 25E, X275, ... |
−50 | 2, 3, 661, E6E, 2325, X565, 16201, 58XE7, ... |
−4E | 15, 37, 6X7, 17551, ... |
−4X | 3, 15, X07, 644E, ... |
−49 | 45, 16E, X657, E85E, 11365, 4291E, 5437E, ... |
−48 | 31, 8E, 747, 23XE, ... |
−47 | 3, 5, 12E, 171, 7X1, 921, 1377, 8871, ... |
−46 | 2, 7, 17, 57, 145, 6X7, 4974E, 62225, 7E487, ... |
−45 | 10847, 12361, 77771, ... |
−44 | 7, 117, 145, 167, 32E, 3081, 26745, 4140E, ... |
−43 | 3, 105, 1X71, ... |
−42 | 801, 1369E, 28905, ... |
−41 | 7, 17, 31, 6E, X35, 72EE, E7E1, ... |
−40 | 2, 5, 15, XE, 40E51, ... |
−3E | 5, 17, 1E, 67, 1047, 4541, ... |
−3X | 7, 1E, 4E, 5E, 8E, 167, 237, 133E, 3E61, 46X75, ... |
−39 | 87, 111, 19607, ... |
−38 | 2, 7, 1EX41, ... |
−37 | 5, 7, 17, 18E, 1E1, 27E, 35E, 18E7, 2745, 5927, 15145, ... |
−36 | 2, 3, 4E1, E45, X447, 61X21, 83E07, ... |
−35 | 15, 497, 559E1, ... |
−34 | 45, 57, 855, 348E, 367E, 6915, 15407, ... |
−33 | 3, 11, 105, 8X95, ... |
−32 | 2, 5, 11E, 747, E11, 1711, 1E51, 7E0E, 4066E, 6419E, ... |
−31 | 5, 7, 1697, 7X535, ... |
−30 | 27, 13E, 195, 267, 1931, 53E47, ... |
−2E | E, 11, 67, X7, 35E, 435, 4E1, 5E5, X4E, 2267, 6659E, ... |
−2X | 3, 122371, ... |
−29 | 5, 57, 111, 7097, 131555, ... |
−28 | 2 (no others) |
−27 | 91, 325, 745, 25475, ... |
−26 | 2, E7, 125, 397, 591, 125E, 16X7, 1971, 5X67, 2887E, 3X787, ... |
−25 | 7, 54XX1, 73581, ... |
−24 | 3, 17, 271, 2XE, 34E, 71E, 403X1, X9331, ... |
−23 | (none) |
−22 | E, 91, 16E, 1E1, 24E, 5E5, 13E5, 5297, ... |
−21 | 3, 7, 1E, 25, 4E, 881, EX5, 107E, 114E, 1EX1, 5167, 21001, 3966E, ... |
−20 | 2, 7, E, 17, 133E, 1525, 2X47, ... |
−1E | E, 11, 57, 91, 237, 40E, 11E1E, 15361, 215E1, ... |
−1X | 3, 5, 11, 37, 67, 85, 8E, 16E, 255, 4341, 25127, ... |
−19 | 3, 5, 7, 11, 31, 24E, X225, 2X2EE, 3X8X1, X1X5E, 170417, ... |
−18 | 2, 5, 67, 75, 4E1, 565, 80E, 404E, 69071, 86EX7, 16E6E5, ... |
−17 | 15, 31, 111, 117, 447, 4307, 1319E, 15935, 1EX15, 3913E, 1E0001, ... |
−16 | 2, 3, 7, 1E, 61, 511, 665, 775, 1151, 2837, 1E2537, ... |
−15 | 7, 15, 1E, 3E, 687, 3X25, 4975, 1EX31, 55905, E32X1, 149657, ... |
−14 | 3, 5, 7, 1E, 31, 75, 105, 125, 18E, 217, 225, 15585, 415495, ... |
−13 | 3, 7, 25, 76E, 149E, 27615, 33081, 22749E, ... |
−12 | 2, 7, 45, 35E, 865, 11125, 447721, ... |
−11 | 3, E, 15, 17, 647, 7EE, 1747, 548E, 28725, 49X177, ... |
−10 | 2, 5, E, 91, 141, X37, 66X1, 1048E, 1081E, 11E1E, 51X0E, 68X4E, 1EE015, ... |
−E | 5, 7, 12E, 171, 307, 3X5, 3655, X9767, 1392X1, 912657, ... |
−X | 5, 7, 17, 27, 45, 57, 205, 455, 12X1, 18XE, 10E267, 65246E, ... |
−9 | 3, 4E, 167, 397, 545, 701, 107E, 224E, 2459E, 939EE, 29E081, 355851, ... |
−8 | 2 (no others) |
−7 | 3, 15, 1E, 25, 3E, 51, E2E, X68E, 5154E, 98845, 489895, ... |
−6 | 2, 3, E, 27, 37, 3E, 4E, 8E, 577, 176E, 2955, 5681, 17525, 1X2EE, 1EE11, 643XE, 95755, 534077, ... |
−5 | 5, 57, 85, 87, 171, 24E, 23X5, 11595, 15531, 86761, 94297, 10X527, 1144X7, 142145, 7561XE, ... |
−4 | 2, 3 (no others) |
−3 | 2, 3, 5, 7, 11, 1E, 37, 1E5, 25E, 347, 401, XE7, E67, 1011, 1X1E, 53E5, 6621, 745E, 76E1, 9X2E, 134E5, 503XE, 65817, 74167, 299741, 4X172E, 7755X7, X224E7, ... |
−2 | 3, 4, 5, 7, E, 11, 15, 17, 1E, 27, 37, 51, 67, 85, X7, 11E, 13E, 147, 221, 24E, 4X5, EX5, 1621, 206E, 343E, 60E1, 622E, 663E, 7207, 8467, 20895, 4028E, 47235, 57X1E, 6161E, 684X1, 69787, 10X635, 1102XE, 160755, 3E6867, 1424E9E, ..., 457817E, 458X897, ... |
2 | 2, 3, 5, 7, 11, 15, 17, 27, 51, 75, 8E, X7, 375, 427, 8X7, 1337, 13X1, 1X41, 2565, 2687, 5735, 5905, 65X5, E655, 10685, 11521, 21901, 41XXE, 53E47, 64501, X5077, 305E9E, 355435, 507X77, 575225, EE6425, 1018595, 2403095, 4615431, 7046577, 8072087, 884201E, X222021, XXE3855, 1053X84E, 12346161, 12531515, ..., 17476435, ..., 20X28041, ..., 21X46E85, ..., 237XE125, ... |
3 | 3, 7, 11, 5E, 87, 391, 76E, 95E, E37, 2501, 526E, 563E, 19441, 20E07, 24901, 29625, 1E3X4E, 364017, 8XX007, ... |
4 | 2 (no others) |
5 | 3, 7, E, 11, 3E, X7, 105, 131, 437, 655, 1E7E, 6405, 77E5, 8041, 9687, 9863E, 1714X5, 770907, ... |
6 | 2, 3, 7, 25, 5E, X7, 1X7, 365, 735, 3845, 3E97, 6185, E615, 3X357, 253XXE, 559E37, ... |
7 | 5, 11, XE, 105, E97, 8291, 18455, 60E31, 15E105, 50EX77, ... |
8 | 3 (no others) |
9 | (none) |
X | 2, 17, 1E, 225, 71E, 244X1, 42045, 53301, 110547, 1E35561, 28X421E, ... |
E | 15, 17, 61, E7, 637, 112E, 1211, 2941, 2E95, 6357, E801, 12205E, 761707, ... |
10 | 2, 3, 5, 17, 81, 91, 225, 255, 4X5, 5777, 879E, 198E1, 23175, 311407, ... |
11 | 5, 7, E5, 1E7, 617, 6X7, 711, 835, 215E, XX1E, 1645E, 4X601, 6060EE, ... |
12 | 3, 7, 17, 27, 35, 167E, E495, 2X665, 33025, 193741, ... |
13 | 3, 37, 61, 347, 15XE, 5085, 19801, 43615, 204391, 26X335, ... |
14 | 2 (no others) |
15 | 3, 5, 7, E, 3E, 5E, 2XE, 293E, 18411, 27947, 37151, 7EEE0E, ... |
16 | 2, 12X2E, 14807, 6X23E, 76X77, 88331, 17E8X5, ... |
17 | 17, 27, 3E, 4E, 51, 8E, 241, 745, 5607, 10917, X11X7, ... |
18 | 3, E, 15, X3E, 15E45, 24337, 2E64E, 1X9685, ... |
19 | 3, E, 15, 37, 1X7, 764X1, 139X81, ... |
1X | 2, 5, 67, 85, 25E, 5E5, 26EE, 5285, 14127, ... |
1E | 5, 1X11, 2E681, 4525E, 5X6X5, X816E, ... |
20 | 3, 5, 17, 45, 5E, 465, 471, 5E9E, 2464E, 56X91, 744EE, ... |
21 | (none) |
22 | 7, 37, 24E, 7231, 7275, 13565, ... |
23 | 3 (no others) |
24 | 2, 5, 15, 321, 9X7, 57085, ... |
25 | 5, 107, 219E, 2458E, 38845, ... |
26 | 2, 5, E, 117, 3E5, 1051, 4X7E, 36207, 39775, 3EE6E, ... |
27 | 7, 15, 27, 3291, 5931, 27651, 4X61E, 219E2E, ... |
28 | (none) |
29 | 3, 145, 20X5, 3E87, 8X351, ... |
2X | 11, X45, 3477, 3837, 60491, ... |
2E | 221, 901, ... |
30 | 2 (no others) |
31 | 11, 5E, 131, 18E, 327, 375, 42X1, 19135, 23X51, 42711, 8163E, 100365, ... |
32 | 3, 7, 295, 315, 53125, ... |
33 | 251, 447, 2725, 9761, 19045, ... |
34 | 2, 5, 7, 17, 1E, 25, 391, 527, 8X5, ... |
35 | 3, 6E, 1X5, 2X1, 1027, 6957, ... |
36 | 2, 91E, ... |
37 | 5, 11, 3771, 135E5, 1396E, 1E1EE, 21XE1, 94407, ... |
38 | 5, 27, 11E, 4X1EE, ... |
39 | 17, 45, 11E, 1E07, 6621, 17X67, X539E, ... |
3X | 2, 7, 17, 57, 157, 301, 14E1, 16X7, E377, ... |
3E | X7, X511, 1XE1E, ... |
40 | 17, 1X5, 251, 27E, 907, 8847, 97EE7, ... |
41 | (none) |
42 | 3, 5, X7, E7, 24E, 471, 1337, 3935, X1867, ... |
43 | 2545, 18477, ... |
44 | 2, 87, 195, 2545, 399E, ... |
45 | E, 27, 35, XXE, 12XE7, 89391, ... |
46 | 3, 285, 9655, X76E, 3E887, ... |
47 | 15, 35, 3E, 107, 59E, 138E, 1E0E, 2127, 3335, 1869E, 48E57, 87E05, ... |
48 | 7, 111, 1257, 1471, 29537, ... |
49 | 3, 15, 91, 107, 157, 471, 9997, 10905, 113645, ... |
4X | 2, 35, 1425, 33325, ... |
4E | 3, 11, 33E, 710E, 82301, ... |
50 | 2, 7, E, 45, 125, ... |
51 | 7, 31, 8E, 541, ... |
52 | 3, 5, 15, 3E, 117, 125, 531, 2787, 5405, 6375, ... |
53 | 5, 1937, 1X415, 95445, ... |
54 | (none) |
55 | 17, 25, 447, ... |
56 | 2, 3, 7, 17, E685, ... |
57 | 17, 267, X3E, 1E2E, 26XE, 601E, 7917, 80XX1, ... |
58 | 5, 7, 8E, 105, 1727, ... |
59 | 3, 51, 1457, 2085, 4971, 516X5, 87091, ... |
5X | 2, 25, 4E, 391, 535, 705, 6885, 13EX7, ... |
5E | 3, 27, 35, 111, XEE, 15E9E, 27X5E, 35837, 73261, 8382E, ... |
60 | 2, 7, 11, 91, 16E, ... |
61 | 5, 7, 185X1, 5400E, ... |
62 | 5, 13E, 1X75, 16117, ... |
63 | 3, 17, 3E, 61, 517, 774E, 9047, 45EE7, ... |
64 | 35, 111, 307, 415, 1E4E, 1E85, 2771, 77295, ... |
65 | 3, 5, 31, 8X81, 98155, ... |
66 | 2, 3, 85, 195, 1165, 32X31, E9X7E, ... |
67 | 5, 91, 105, 46E, 14691, ... |
68 | 3, 7, ... |
69 | (none) |
6X | 2, 1E, 27, 35, 449E, 7607, ... |
6E | 5, 16X1, ... |
70 | 15, 2325, ... |
71 | 5, 17, 127E, ... |
72 | E, 37, 95, 365, 751, 1825, 2607, 1E59E, ... |
73 | 7, 15, 5X37E, 105161, ... |
74 | 2, 51, 401, 21X7, 1124E, 1E6EE, ... |
75 | 3, 7, 37, 3E, 5E, 91, 3E7, 6E17, 24E85, ... |
76 | 3, 17, 81, 3021, 78347, ... |
77 | 2685, E7E1, ... |
78 | 307, 7635, 11151, 218E7, ... |
79 | 7, 2X07, ... |
7X | 5, 11, 31, 1051, 20X5, 82717, E8061, ... |
7E | 7, 377, 5457, 609E, 678E, ... |
80 | 2, 1E27, 23127, ... |
81 | 15, 31, E91, 90541, ... |
82 | 11, 3E, 1755, ... |
83 | 3, 5, 31, 3E, 27E, 3277, ... |
84 | 2 (no others) |
85 | 3, 241, 485, 825, 399E, ... |
86 | 2, 4E, 481, 12627, ... |
87 | 17, 221, X91, ... |
88 | 81, 19E, 3191, ... |
89 | 3, 17, 285, 167E, 2927, ... |
8X | 2, 105, ... |
8E | 15, 1204E, ... |
90 | 2, 315, 1525, ... |
91 | 15, 835, 7XEE, 137E1, ... |
92 | 3, 5, 11, 497, EE5, 1E01, 6X17, ... |
93 | 3, 241, ... |
94 | 2, 67, 8E, 4X5, E95, 3335, ... |
95 | 1E, 31, 396E, ... |
96 | 25, 37, 61, 75, 3E5, 4E1, ... |
97 | 7, 181, 995, 1431, 1577, 4535, 730E, 9927, ... |
98 | 4E, 1547, ... |
99 | 3, 5, 17, 27, ... |
9X | 5, 117, 141, ... |
9E | 3, 17, 58E, 136E, 2265, ... |
X0 | 5, 271, E91, ... |
X1 | (none) |
X2 | 5, 7, 57, 224E, ... |
X3 | 37, 3XE, E91, 29X5, 111E1, ... |
X4 | 41E, X767, 14667, ... |
X5 | (none) |
X6 | 2, 7, 31, 4E, X7, 10157, ... |
X7 | 5, 1E, 27, 11E, 3081, 5235, 11595, 7EX01, ... |
X8 | 7 (no others) |
X9 | 5, 15, 91, 4X7E, ... |
XX | 2, 31, ... |
XE | 3, 27, 19E, ... |
E0 | 3E, 5E, 1E27, ... |
E1 | 11, 41E, 6X7, 825, 194E, 86E7, ... |
E2 | 5, 31, 255, 178E, 10457, ... |
E3 | 817, 898E, ... |
E4 | 2, 16E, 205, 2485, ... |
E5 | E, 17, 701, 184E, ... |
E6 | 2, 3, 51, 7XEE, ... |
E7 | 117, 125, 2265, ... |
E8 | 67, 401, EE5, ... |
E9 | 3, 1E, 125, 1X41, ... |
EX | 867, 3671, ... |
EE | 3, 5, ... |
100 | (none) |
prime p | positive bases such that Rp is prime | negative bases such that Rp is prime |
3 | 2, 3, 5, 6, 8, 10, 12, 13, 15, 18, 19, 20, 23, 29, 32, 35, 42, 46, 49, 4E, 52, 56, 59, 5E, 63, 65, 66, 68, 75, 76, 83, 85, 89, 92, 93, 99, 9E, XE, E6, E9, EE, 103, 106, 109, 10E, 115, 116, 118, 11E, 120, 125, 128, 138, 139, 140, 142, 14E, 152, 155, 15E, 162, 173, 178, 185, 186, 1X2, 1X8, 1E2, 1E3, 1EE, 200, 202, 205, 219, 222, 235, 238, 240, 246, 248, 250, 253, 259, 269, 276, 279, 27E, 288, 28E, 292, 296, 298, 299, 2X6, 2X8, 2E6, 302, 303, 313, 319, 31E, 320, 338, 349, 353, 358, 368, 372, 379, 382, 383, 385, 389, 390, 39E, 3X2, 3X8, 3E2, 3E3, 3E8, 403, 406, 408, 423, 425, 429, 430, 439, 440, 442, 44E, 456, 458, 478, 47E, 485, 492, 4X0, 4X5, 500, 506, 508, 513, 51E, 523, 52E, 535, 536, 540, 545, 552, 555, 560, 566, 569, 578, 582, 583, 589, 58E, 598, 59E, 5X6, 5E3, 5E5, 5E8, 600, 60E, 612, 622, 626, 629, 62E, 63E, 643, 646, 648, 653, 672, 67E, 680, 689, 692, 699, 6X3, 6X6, 6X8, 6X9, 6E5, 6E6, 6EE, 703, 708, 710, 712, 719, 729, 742, 749, 74E, 752, 756, 759, 76E, 770, 775, 776, 783, 790, 792, 799, 7X2, 7X5, 7E2, 7E6, 7E8, 7E9, 80E, 825, 826, 833, 835, 838, 840, 842, 849, 855, 859, 866, 886, 888, 890, 893, 8X0, 8X2, 8X3, 8X9, 8EE, 910, 916, 919, 923, 92E, 938, 93E, 945, 948, 953, 958, 959, 965, 970, 976, 980, 989, 99E, 9X5, 9X6, 9XE, 9E6, X00, X02, X03, X08, X09, X12, X29, X2E, X38, X3E, X46, X55, X63, X68, X72, X75, X80, X83, X8E, X90, X9E, XX0, XX2, XX6, XX8, XE9, XEE, E06, E08, E10, E23, E28, E30, E40, E50, E52, E56, E62, E63, E69, E78, E85, E90, E93, E98, E99, EX3, EX5, EE0, EE2, ... | 2, 3, 4, 6, 7, 9, 11, 13, 14, 16, 19, 1X, 21, 24, 2X, 33, 36, 43, 47, 4X, 50, 53, 57, 5X, 60, 64, 66, 67, 69, 76, 77, 84, 86, 8X, 93, 94, 9X, X0, E0, E7, EX, 100, 104, 107, 10X, 110, 116, 117, 119, 120, 121, 126, 129, 139, 13X, 141, 143, 150, 153, 156, 160, 163, 174, 179, 186, 187, 1X3, 1X9, 1E3, 1E4, 200, 201, 203, 206, 21X, 223, 236, 239, 241, 247, 249, 251, 254, 25X, 26X, 277, 27X, 280, 289, 290, 293, 297, 299, 29X, 2X7, 2X9, 2E7, 303, 304, 314, 31X, 320, 321, 339, 34X, 354, 359, 369, 373, 37X, 383, 384, 386, 38X, 391, 3X0, 3X3, 3X9, 3E3, 3E4, 3E9, 404, 407, 409, 424, 426, 42X, 431, 43X, 441, 443, 450, 457, 459, 479, 480, 486, 493, 4X1, 4X6, 501, 507, 509, 514, 520, 524, 530, 536, 537, 541, 546, 553, 556, 561, 567, 56X, 579, 583, 584, 58X, 590, 599, 5X0, 5X7, 5E4, 5E6, 5E9, 601, 610, 613, 623, 627, 62X, 630, 640, 644, 647, 649, 654, 673, 680, 681, 68X, 693, 69X, 6X4, 6X7, 6X9, 6XX, 6E6, 6E7, 700, 704, 709, 711, 713, 71X, 72X, 743, 74X, 750, 753, 757, 75X, 770, 771, 776, 777, 784, 791, 793, 79X, 7X3, 7X6, 7E3, 7E7, 7E9, 7EX, 810, 826, 827, 834, 836, 839, 841, 843, 84X, 856, 85X, 867, 887, 889, 891, 894, 8X1, 8X3, 8X4, 8XX, 900, 911, 917, 91X, 924, 930, 939, 940, 946, 949, 954, 959, 95X, 966, 971, 977, 981, 98X, 9X0, 9X6, 9X7, 9E0, 9E7, X01, X03, X04, X09, X0X, X13, X2X, X30, X39, X40, X47, X56, X64, X69, X73, X76, X81, X84, X90, X91, XX0, XX1, XX3, XX7, XX9, XEX, E00, E07, E09, E11, E24, E29, E31, E41, E51, E53, E57, E63, E64, E6X, E79, E86, E91, E94, E99, E9X, EX4, EX6, EE1, EE3, ... |
5 | 2, 7, 10, 11, 15, 1X, 1E, 20, 24, 25, 26, 34, 37, 38, 42, 52, 53, 58, 61, 62, 65, 67, 6E, 71, 7X, 83, 92, 99, 9X, X0, X2, X7, X9, E2, EE, 101, 10X, 116, 118, 119, 124, 127, 129, 141, 146, 150, 154, 166, 16E, 17E, 189, 192, 193, 198, 19E, 1X1, 1XX, 1XE, 1E1, 1E9, 200, 204, 214, 218, 233, 241, 244, 254, 25E, 269, 271, 289, 291, 2X0, 2X6, 2X9, 2XX, 305, 307, 314, 321, 323, 336, 33E, 34X, 350, 353, 35E, 361, 36X, 373, 380, 382, 38X, 38E, 390, 39X, 3X5, 3XE, 3E3, 3E4, 3E8, 403, 416, 424, 435, 451, 459, 462, 472, 487, 49X, 4X3, 4E6, 508, 521, 526, 539, 542, 544, 561, 570, 571, 588, 599, 5E2, 5E8, 600, 604, 614, 622, 647, 666, 676, 687, 688, 692, 697, 6EE, 701, 730, 731, 733, 739, 75X, 75E, 765, 77X, 784, 7X1, 7X5, 7X7, 800, 802, 805, 819, 822, 827, 82E, 849, 864, 873, 881, 885, 8XX, 8E0, 901, 927, 931, 93E, 944, 948, 962, 969, 972, 986, 9X3, 9E1, X10, X1X, X1E, X2E, X34, X45, X53, X68, X71, X95, X99, XX2, XX3, XE7, E61, E66, E67, E70, E74, E7E, E83, E84, E92, E93, EX0, ... | 2, 3, 5, X, E, 10, 14, 18, 19, 1X, 29, 31, 32, 37, 3E, 40, 47, 5E, 63, 64, 68, 69, 93, X1, X6, XE, E1, E3, E4, E9, 10E, 111, 112, 119, 128, 129, 130, 14E, 167, 182, 185, 18E, 191, 194, 195, 196, 1X1, 1X4, 1E0, 1EX, 202, 209, 217, 22X, 22E, 246, 261, 263, 266, 273, 275, 281, 284, 290, 2X3, 2E7, 306, 318, 326, 343, 347, 353, 358, 361, 362, 363, 366, 380, 383, 39E, 3X3, 407, 410, 423, 426, 42X, 444, 455, 456, 464, 46X, 484, 499, 4X6, 4XE, 4E8, 516, 519, 532, 534, 540, 552, 55E, 560, 578, 579, 57E, 598, 5XX, 5E9, 631, 640, 65E, 680, 691, 696, 698, 69X, 704, 713, 716, 723, 731, 741, 749, 74X, 750, 753, 767, 771, 781, 790, 79E, 7X9, 7E8, 803, 804, 806, 808, 811, 812, 822, 82E, 833, 853, 866, 877, 895, 896, 89X, 8X4, 8X9, 8E7, 912, 920, 939, 93X, 949, 959, 960, 967, 989, 9E5, X10, X1X, X29, X2E, X40, X48, X56, X76, X85, X92, X93, XX0, XX7, E01, E09, E12, E19, E23, E27, E2X, E31, E45, E4E, E64, E65, E78, E86, E90, E93, E96, EXE, EE0, EE1, ... |
7 | 2, 3, 5, 6, 11, 12, 15, 22, 27, 32, 34, 3X, 48, 50, 51, 56, 58, 60, 61, 68, 73, 75, 79, 7E, 97, X2, X6, X8, 102, 105, 110, 112, 114, 117, 130, 136, 140, 14E, 152, 154, 164, 165, 177, 179, 17X, 18E, 1X0, 1X2, 1E4, 1E6, 202, 206, 210, 213, 228, 245, 251, 252, 255, 263, 279, 28E, 293, 298, 299, 2X9, 2XX, 304, 306, 313, 316, 325, 328, 32X, 33X, 377, 383, 38E, 398, 3X8, 407, 408, 413, 41E, 429, 42E, 43X, 45X, 45E, 467, 469, 470, 489, 4X2, 4X4, 4E2, 4E9, 512, 534, 539, 548, 566, 568, 56X, 571, 586, 5X2, 5E4, 5EE, 606, 612, 62E, 640, 641, 652, 653, 657, 664, 665, 666, 66E, 67E, 6X0, 6E2, 719, 732, 738, 73E, 761, 764, 767, 787, 78E, 791, 7X3, 7X8, 7E4, 7E8, 809, 815, 830, 831, 836, 837, 839, 859, 860, 865, 880, 896, 8X5, 8X8, 8X9, 8E0, 8E7, 901, 911, 918, 931, 937, 944, 957, 958, 963, 969, 972, 98X, 990, 995, 9X2, 9X3, 9X8, 9X9, 9EX, 9EE, X13, X25, X34, X36, X41, X46, X60, X75, X81, X8E, X92, X96, X9X, XX3, XX5, XX6, XX7, XE0, XE7, E01, E10, E19, E1X, E20, E21, E32, E37, E3X, E57, E70, E74, E7E, E87, E98, EX4, EX6, EXX, ... | 2, 3, X, E, 12, 13, 14, 15, 16, 19, 20, 21, 25, 31, 37, 38, 3X, 41, 44, 46, 51, 56, 60, 61, 66, 70, 72, 73, 83, 85, 8X, 96, 97, X8, E1, E3, E4, EE, 101, 104, 118, 121, 122, 125, 148, 163, 168, 16X, 170, 173, 176, 180, 188, 193, 19X, 1XE, 1E5, 1E6, 20X, 211, 215, 228, 22X, 233, 236, 243, 24E, 25E, 268, 269, 281, 283, 295, 2X9, 303, 30X, 316, 31E, 322, 348, 352, 364, 373, 375, 3X1, 403, 406, 412, 434, 439, 441, 444, 446, 452, 466, 479, 483, 497, 49X, 4E3, 4E4, 4E6, 4EE, 504, 505, 508, 510, 515, 526, 530, 531, 539, 540, 545, 546, 54X, 550, 561, 56E, 576, 58E, 590, 593, 5X2, 5X8, 601, 605, 631, 63X, 647, 64X, 652, 657, 660, 662, 680, 687, 68E, 69X, 6E6, 702, 712, 720, 744, 755, 756, 769, 785, 786, 797, 7X0, 7X2, 7X3, 7X4, 7XX, 808, 822, 823, 828, 832, 847, 854, 856, 858, 859, 8E7, 8E9, 906, 907, 930, 945, 949, 955, 956, 95X, 965, 966, 976, 982, 993, 9E7, X2X, X3E, X63, X70, X7E, X82, X9X, XX2, XX3, XX4, E11, E19, E26, E27, E36, E46, E47, E51, E54, E59, E5X, E73, E81, E83, E94, E96, EX1, EX6, EX9, EE3, EE7, EEX, ... |
E | 5, 15, 18, 19, 26, 45, 50, 72, E5, 124, 143, 158, 168, 171, 196, 1X4, 1X8, 227, 243, 257, 265, 266, 285, 286, 292, 2X6, 32E, 340, 360, 386, 38E, 393, 3E3, 414, 437, 442, 466, 4E1, 513, 530, 53X, 542, 54X, 557, 572, 578, 60X, 667, 691, 6E7, 724, 728, 749, 76E, 77E, 836, 84X, 851, 873, 877, 881, 894, 911, 913, 939, 944, 951, 969, 9X9, 9E5, 9E9, X08, X09, X17, X5X, X74, XE3, E02, E15, E3E, E65, E94, E96, E99, ... | 2, 6, 10, 11, 1E, 20, 22, 2E, 52, 59, 77, 89, 103, 114, 117, 133, 135, 141, 171, 174, 175, 178, 188, 19X, 1X5, 1E4, 206, 213, 223, 236, 274, 28X, 2E6, 2EX, 308, 328, 331, 34E, 350, 364, 389, 441, 445, 45E, 465, 476, 528, 544, 547, 556, 558, 594, 5X0, 5E0, 5E2, 5E3, 5E7, 635, 64X, 690, 6X9, 6E2, 704, 729, 72E, 745, 758, 762, 7X1, 7X8, 806, 819, 821, 83E, 848, 871, 896, 8X5, 914, 934, 936, 962, 976, 991, 9EE, X02, X13, X58, X6X, X87, X95, XX4, XX6, E36, E69, ... |
11 | 2, 3, 5, 7, 2X, 31, 37, 4E, 60, 7X, 82, 92, E1, 105, 107, 113, 13X, 153, 163, 165, 18E, 198, 1X0, 1X3, 1E6, 1EX, 203, 227, 257, 263, 271, 27X, 291, 292, 296, 29X, 2X0, 2X4, 304, 30X, 346, 349, 363, 392, 394, 3X0, 3X1, 406, 409, 414, 423, 42X, 432, 44X, 457, 459, 495, 4E0, 500, 50X, 520, 541, 544, 572, 5XE, 5E1, 5EX, 616, 640, 654, 656, 674, 688, 6X9, 6EX, 70E, 715, 728, 733, 775, 776, 794, 799, 7X3, 7X5, 801, 811, 818, 81E, 833, 879, 880, 89E, 8X1, 8X9, 903, 909, 90X, 913, 942, 968, 96X, 988, 98E, 990, 998, 9E0, X02, X0X, X27, X52, XXX, E01, E64, E66, E6X, ... | 2, 3, 19, 1X, 1E, 2E, 33, 62, 68, 70, 75, 90, 96, X1, X6, E2, 108, 109, 123, 130, 143, 150, 172, 194, 19E, 209, 229, 232, 240, 246, 249, 253, 268, 286, 289, 291, 292, 293, 2X5, 2E7, 307, 34X, 379, 3XE, 3E2, 3EX, 413, 422, 429, 446, 455, 464, 469, 470, 477, 48X, 491, 49X, 4E0, 4E4, 4E8, 501, 52E, 534, 537, 55E, 565, 579, 590, 59E, 5XX, 5EX, 616, 62X, 6E3, 715, 719, 71X, 723, 726, 735, 747, 755, 758, 761, 781, 787, 7X0, 80X, 820, 822, 829, 854, 863, 872, 894, 898, 8X2, 90E, 91X, 924, 953, 961, 963, 965, 978, 980, X05, X0X, X0E, X19, X41, X48, X75, X83, X92, X97, XX0, E22, E24, E28, E40, E93, ... |
15 | 2, E, 18, 19, 24, 27, 47, 49, 52, 70, 73, 81, 8E, 91, X9, 103, 105, 111, 114, 122, 131, 139, 13E, 153, 181, 187, 18E, 1XX, 207, 209, 223, 233, 23E, 251, 253, 257, 264, 265, 268, 277, 27E, 2X2, 2XE, 2E3, 2EE, 304, 306, 32X, 334, 362, 369, 37E, 3X5, 3E7, 419, 41E, 432, 451, 465, 468, 494, 4E0, 4E1, 500, 518, 536, 597, 598, 60X, 692, 694, 698, 69X, 6X2, 719, 731, 73X, 7X8, 7EE, 817, 825, 829, 85X, 866, 870, 874, 8X8, 91E, 930, 95E, 963, 967, 980, 98X, 9E7, X07, X35, X47, X48, X67, X71, X76, X80, X8E, X98, E08, E17, E1E, E27, E83, E90, EX2, EEX, ... | 2, 7, 11, 15, 17, 35, 40, 4X, 4E, 56, 72, X9, E1, E7, EE, 102, 105, 11X, 11E, 173, 1X4, 1X6, 20E, 234, 25E, 283, 291, 2X0, 331, 377, 37E, 386, 391, 3X1, 3X3, 3E5, 406, 407, 420, 428, 44X, 474, 477, 480, 481, 4E1, 4E6, 500, 505, 511, 522, 533, 548, 56X, 574, 586, 5E8, 606, 611, 624, 628, 632, 63E, 682, 68X, 693, 722, 736, 737, 762, 76X, 785, 7E8, 7EX, 814, 829, 83X, 840, 849, 850, 869, 86E, 891, 8X9, 914, 936, 937, 951, 970, 972, 977, 98E, 997, X00, X01, X17, X31, X37, X49, X53, XX0, XX1, XE6, XEE, E89, E8E, EX6, ... |
17 | 2, X, E, 10, 12, 17, 20, 34, 39, 3X, 40, 55, 56, 57, 63, 71, 76, 87, 89, 99, 9E, E5, 103, 118, 11E, 12E, 131, 151, 164, 177, 182, 191, 192, 19E, 1X4, 1E1, 213, 223, 238, 241, 266, 269, 26X, 285, 293, 298, 2E4, 2EE, 312, 315, 340, 341, 362, 365, 375, 377, 383, 397, 3E3, 3E9, 405, 43X, 45X, 463, 481, 514, 540, 557, 565, 573, 576, 577, 581, 5X0, 5X6, 5E5, 603, 605, 606, 620, 62E, 632, 68E, 6X4, 6X6, 6X8, 702, 713, 71X, 729, 72X, 754, 77X, 788, 790, 7E3, 7E7, 848, 856, 85E, 884, 888, 889, 890, 8E7, 92X, 933, 938, 98X, X12, X80, X82, E0X, E32, E35, E3E, E4X, E54, E91, E99, EX5, EEX, ... | 2, X, 11, 20, 24, 37, 3E, 41, 46, 55, 6E, 82, 8X, EE, 108, 129, 134, 13X, 142, 143, 181, 189, 197, 1X0, 204, 223, 227, 249, 255, 257, 282, 28X, 341, 358, 371, 379, 386, 387, 3X4, 417, 421, 461, 468, 488, 492, 493, 4X1, 4XE, 4E2, 530, 566, 570, 581, 582, 591, 59E, 5X1, 5E6, 600, 623, 636, 640, 658, 670, 691, 6X7, 6XX, 704, 71X, 720, 724, 732, 741, 744, 74X, 753, 769, 812, 81E, 843, 845, 863, 871, 891, 8XE, 8E0, 91E, 927, 933, 934, 935, 950, 964, 977, 998, 999, 9EE, X0X, X18, X37, X40, X47, X52, X63, X66, X69, X78, E78, E79, E85, E86, EE4, EE6, ... |
1E | X, 34, 6X, 95, X7, E9, 122, 195, 1XE, 1EE, 207, 223, 248, 271, 30X, 330, 429, 44X, 45X, 473, 47E, 4E2, 583, 596, 5E5, 618, 626, 634, 6X8, 6E1, 744, 751, 759, 794, 7E3, 832, 868, 885, 891, 8X8, 8E7, 943, 993, 9E8, X08, X37, X62, XXX, E4X, E84, E8E, ... | 2, 3, 7, 14, 15, 16, 21, 3X, 3E, 8X, 92, 93, 9X, E4, 100, 101, 172, 17X, 261, 27X, 2E2, 307, 336, 352, 353, 373, 410, 453, 469, 4XE, 511, 527, 56X, 593, 5EE, 632, 651, 694, 6XX, 6E5, 743, 7E0, 825, 874, 881, 91E, 967, X02, X16, X58, E97, ... |
25 | 6, 34, 55, 5X, 96, 107, 165, 171, 1X4, 1E7, 292, 317, 324, 373, 3X2, 40E, 443, 444, 46E, 493, 4X3, 5E7, 618, 643, 667, 692, 6X2, 779, 794, 807, 820, 850, 85E, 901, 914, 922, 935, 97X, 995, 99E, X06, X31, X58, E08, E44, E8E, ... | 7, 13, 21, 52, 9E, X3, 10X, 185, 1E9, 206, 207, 259, 26E, 338, 442, 474, 496, 4E0, 503, 515, 540, 553, 56E, 589, 58X, 597, 5X1, 5X2, 603, 607, 629, 634, 65X, 675, 74X, 751, 768, 775, 784, 827, 839, 842, 88E, 906, 958, X19, X49, X74, X9E, E46, EXE, ... |
27 | 2, 12, 17, 27, 38, 45, 5E, 6X, 99, X7, XE, 101, 129, 145, 14E, 181, 196, 199, 1E0, 1E7, 205, 228, 231, 277, 283, 284, 29X, 2X5, 325, 326, 332, 346, 34E, 386, 399, 3E5, 406, 430, 436, 453, 4X0, 4XX, 503, 522, 539, 53E, 546, 564, 56X, 611, 612, 633, 64E, 699, 6X7, 6E2, 6E6, 700, 735, 759, 789, 78X, 78E, 7XX, 7E6, 807, 821, 831, 840, 865, 89X, 8X9, 914, 953, 9E1, X3X, X45, X48, X9X, E19, E26, E36, E40, E55, E75, E96, ... | 2, 6, X, 30, 55, 62, 66, 6E, 8X, 97, X0, 104, 115, 117, 120, 128, 139, 142, 145, 1X2, 1X6, 200, 237, 281, 293, 29E, 2X2, 2X4, 2X5, 2EE, 330, 369, 378, 396, 3E5, 3E8, 402, 405, 420, 42E, 441, 442, 45E, 4EE, 503, 530, 594, 596, 5X9, 64X, 6X2, 6EX, 6EE, 733, 748, 77X, 83E, 855, 87E, 933, 945, 949, 964, 966, 972, 988, 991, 996, 9X2, 9X3, 9X6, 9XX, X04, X15, X96, XX8, E06, E32, E7X, E80, E83, E84, E93, EE9, ... |
31 | 51, 65, 7X, 81, 83, 95, X6, XX, E2, 103, 115, 124, 137, 14X, 154, 186, 199, 1X9, 1E9, 212, 228, 300, 310, 35E, 377, 379, 3XE, 476, 488, 4E1, 518, 531, 557, 632, 682, 684, 689, 705, 785, 811, 82X, 857, 947, X74, E03, E64, E96, EX6, ... | 14, 17, 19, 41, 48, 53, 5E, 62, 65, 6E, 78, 80, 83, 114, 124, 145, 146, 172, 181, 1E4, 1E7, 2X7, 2E5, 314, 34X, 374, 443, 497, 513, 554, 56E, 591, 5X2, 5E1, 607, 608, 625, 626, 691, 699, 6X5, 72E, 744, 745, 753, 823, 826, 843, 8E6, 903, 919, 962, 979, X31, X32, X48, XX2, E15, E34, EXX, ... |
35 | 12, 45, 47, 4X, 5E, 64, 6X, 157, 188, 1X7, 208, 224, 2EX, 301, 307, 334, 395, 3X1, 3X3, 418, 473, 485, 48X, 522, 57X, 594, 619, 652, 66E, 67E, 734, 795, 7E6, 81E, 847, 904, 906, 97X, 990, X05, X22, X85, E56, E97, E9X, EX2, ... | 51, 53, 83, 100, 172, 220, 260, 295, 2X5, 2E4, 317, 36E, 392, 3E3, 42X, 436, 43X, 463, 496, 510, 581, 607, 6EE, 778, 804, 8X6, 98E, 9XE, X18, X7E, XE0, E18, E81, ... |
37 | 13, 19, 22, 72, 75, 96, X3, 117, 130, 21X, 238, 275, 2X1, 306, 314, 321, 339, 37X, 386, 3X4, 40X, 430, 465, 475, 496, 498, 4E1, 534, 553, 56E, 585, 5X8, 611, 62E, 639, 666, 699, 729, 72X, 72E, 74X, 750, 751, 782, 79X, 7XE, 839, 912, 976, X30, XX1, E13, E1X, E44, ... | 2, 3, 6, 1X, 4E, 6E, 77, 7E, X0, 104, 143, 144, 149, 187, 190, 1X0, 223, 260, 276, 322, 3X3, 488, 560, 625, 715, 768, 806, 817, 874, 8X8, 9X0, 9E9, E40, E65, E79, E81, ... |
3E | 5, 15, 17, 47, 52, 63, 75, 82, 83, E0, 124, 136, 145, 164, 1X4, 1E2, 1E3, 200, 307, 30E, 354, 403, 407, 40E, 51X, 549, 589, 63E, 686, 6E7, 719, 803, 842, 897, 969, X19, X28, X29, X76, XX2, E42, E82, EX1, ... | 6, 7, 15, 76, X6, E7, EE, 195, 205, 207, 227, 283, 342, 373, 423, 500, 583, 618, 628, 633, 635, 649, 674, 676, 678, 72X, 742, 7E7, 816, 871, 885, 8X7, 8EE, 93E, 95X, 962, 992, X21, X62, E09, E0E, E1E, E21, E32, E99, EX1, EX5, ... |
45 | 20, 39, 50, 119, 177, 1X8, 1E9, 20X, 217, 279, 2X8, 2E9, 43E, 567, 5E6, 650, 655, 660, 722, 74E, 758, 7X2, 85E, 877, 927, X40, E08, E0X, E63, E75, E78, E7X, ... | X, 12, 34, 49, 93, 9E, 29X, 313, 337, 438, 45X, 59E, 5X8, 602, 639, 6E1, 766, 773, 824, 91X, 91E, 930, 9X2, X07, X28, X7X, E29, E81, E84, ... |
4E | 17, 5X, 86, 98, X6, 138, 155, 195, 206, 25E, 317, 325, 330, 332, 452, 465, 4E2, 536, 53X, 551, 588, 631, 663, 684, 6XE, 724, 733, 776, 7EX, 84E, 86X, 877, 888, 8X6, 979, 989, 9X6, 9E8, X12, X44, X6E, E47, E9E, EX1, ... | 6, 9, 21, 3X, 75, 78, 91, E1, E4, E8, 11E, 125, 159, 17E, 193, 1E1, 241, 252, 25E, 3X1, 3X2, 40X, 416, 41E, 453, 498, 4E2, 57E, 5E8, 600, 707, 71X, 736, 772, 782, 786, 788, 837, 83E, 84E, 854, 890, 943, 957, 962, X17, X30, X31, X57, X85, X98, E11, E13, E30, E44, E66, E94, EE8, ... |
51 | 2, 17, 59, 74, E6, 10E, 151, 176, 240, 2E0, 2E5, 31E, 332, 379, 3X3, 3X9, 428, 442, 477, 482, 53X, 54E, 5X6, 5XE, 661, 68E, 6E2, 739, 73E, 788, 7X9, 893, 929, 973, X05, X08, X4E, X88, X93, XE6, E48, E89, EE8, ... | 2, 7, 5X, 12X, 154, 18E, 1XX, 1E0, 202, 232, 234, 252, 2X5, 318, 3X0, 3X6, 416, 417, 425, 427, 442, 557, 55E, 569, 635, 68E, 70E, 76E, 791, 7E0, 83X, 875, 8X2, 951, 9E0, X1X, E18, E53, E5X, EX1, EE1, ... |
57 | 3X, X2, 17X, 214, 222, 223, 234, 238, 24X, 270, 27X, 2E6, 308, 34E, 354, 366, 378, 380, 3E2, 452, 511, 58X, 708, 73X, 753, 852, 911, 936, 978, 989, X15, X60, X80, XEX, E34, EE2, ... | 5, X, 1E, 29, 34, 46, 141, 232, 236, 264, 273, 27X, 284, 298, 306, 315, 3XX, 441, 442, 48E, 4X4, 539, 565, 573, 6E5, 709, 7X8, 825, 854, 876, 881, 886, 8E2, 976, 998, X36, XE9, E60, ... |
5E | 3, 6, 15, 20, 31, 75, E0, 272, 283, 296, 2E1, 303, 319, 328, 34X, 370, 4E0, 514, 647, 66E, 699, 721, 74E, 816, 837, 885, 8E0, 945, 975, 995, X33, EX5, EE8, EEX, ... | 3X, 7X, 83, 139, 16X, 178, 184, 270, 283, 286, 2X1, 2X2, 2E4, 30X, 33X, 390, 3EX, 428, 42E, 458, 465, 49E, 4XE, 5X6, 604, 724, 880, 934, 966, 983, 9X2, 9X8, X91, E69, ... |
61 | E, 13, 63, 96, 143, 15E, 207, 23E, 276, 3X7, 3E2, 462, 470, 594, 607, 634, 686, 711, 788, 843, 932, 9E5, X39, EE8, ... | 16, 15X, 1E4, 28X, 2E2, 310, 313, 3E7, 521, 557, 564, 61X, 676, 680, 6X3, 704, 73E, 7E0, 836, 850, 860, 863, 945, XEE, E12, E48, E86, EXX, EE7, ... |
67 | 1X, 94, E8, 112, 122, 192, 1X7, 236, 23X, 256, 286, 343, 380, 3X8, 3E1, 4E6, 5XX, 620, 650, 655, 659, 65E, 68X, 70E, 733, 7E1, 896, 936, 953, 968, 998, X39, X54, X86, E12, E1E, E2X, ... | 2, 18, 1X, 2E, 3E, 60, 91, E1, 134, 157, 16X, 197, 254, 332, 3X7, 500, 555, 568, 587, 5X2, 627, 6E7, 7E5, 818, 851, 8E7, 91X, 967, X01, X11, X18, X19, X23, X31, E18, E20, E45, E5X, E93, E9E, ... |
6E | 35, 102, 282, 415, 477, 494, 636, 653, 656, 715, 720, 787, 996, 9E9, E46, ... | 41, 63, 322, 333, 44X, 512, 566, 575, 658, 7XE, 898, 988, X4X, ... |
75 | 2, 96, 113, 13X, 176, 18E, 304, 434, 596, 612, 700, 735, 744, 746, 792, X28, E89, ... | 14, 18, 79, 143, 16E, 231, 329, 532, 620, 63E, 757, 905, E06, E48, ... |
81 | 10, 76, 88, 176, 1X7, 243, 2E0, 2E1, 2E8, 2E9, 328, 571, 639, 65X, 73X, 78X, 790, 7X0, 7E4, 807, 835, 958, ... | 5X, X1, 210, 225, 250, 298, 2X4, 324, 36E, 425, 59E, 5X3, 634, 675, 7X2, 7E8, 93X, X22, E76, ... |
85 | 1X, 66, 118, 212, 238, 25E, 283, 2E8, 320, 3E0, 435, 4X1, 4X7, 4X8, 555, 593, 697, 6E7, 731, 927, 978, 988, X55, E02, ... | 2, 5, 1X, 82, 92, XE, E0, E5, 145, 152, 158, 239, 255, 26E, 270, 309, 315, 358, 365, 46X, 4X7, 4E3, 524, 556, 651, 681, 68E, 715, 855, 8X6, 902, 972, 99X, X5E, E74, ... |
87 | 3, 44, 249, 288, 2E1, 334, 408, 435, 449, 535, 53E, 547, 555, 59E, 721, X48, E0X, E64, E75, E90, E92, ... | 5, 39, 80, 8E, E6, EE, 118, 167, 189, 1X8, 243, 2E5, 319, 355, 362, 421, 4X7, 517, 526, 550, 5E7, 658, 666, 702, 725, 819, 8X2, 8E7, 951, 987, X36, X43, X94, E16, ... |
8E | 2, 17, 51, 58, 94, 111, 163, 251, 485, 498, 4X4, 575, 587, 603, 6E3, 821, 84X, 871, 991, X27, X78, XE5, XE6, E64, E68, EXE, ... | 6, 1X, 3X, 48, 67, 259, 275, 322, 331, 454, 486, 5X9, 698, 748, 782, 804, 836, 848, 936, X09, X37, E6X, ... |
91 | 10, 49, 60, 67, 75, X9, 112, 119, 17E, 180, 198, 1E1, 221, 246, 2E1, 311, 401, 669, 6E7, 796, 83X, 860, 898, 971, 987, 9E0, X54, X56, XX5, XX7, E46, E76, ... | 10, 1E, 22, 27, 62, 9X, 121, 140, 199, 35E, 3E0, 3E4, 422, 467, 481, 505, 60X, 619, 689, 690, 6XE, 775, 78X, 834, 841, 857, 889, XX8, E68, E86, ... |
95 | 72, 175, 1X2, 20E, 23X, 350, 414, 455, 468, 4EE, 66X, 990, X31, X5E, E16, E43, E71, ... | 78, 98, 335, 450, 533, 586, 607, 69X, 6E2, 729, XE4, ... |
X7 | 2, 5, 6, 3E, 42, X6, 107, 16X, 18X, 295, 2E7, 335, 339, 346, 355, 409, 440, 458, 486, 491, 493, 532, 628, 629, 660, 758, 987, X59, E8X, ... | 2, 2E, 239, 268, 287, 2E2, 304, 3X2, 455, 482, 501, 5X7, 60X, 951, X64, E18, E74, ... |
XE | 7, 351, 3E3, 413, 415, 431, 538, 617, 62E, 647, 675, 857, 8XE, 922, 93E, 960, ... | 40, 80, 1E5, 2E0, 401, 572, 73E, 795, 798, 9X3, ... |
E5 | 11, 11X, 159, 257, 40X, 479, 4XE, 540, 595, 6E7, 781, 831, 842, 916, 973, X72, E0X, E48, ... | 75, 182, 1EE, 267, 37E, 41E, 489, 50E, 563, 56X, 754, 80X, 929, X98, E05, E24, ... |
E7 | E, 42, 165, 2X7, 375, 401, 404, 478, 4E9, 500, 516, 632, 979, 980, XE9, E56, ... | 26, 70, 1E9, 206, 3X2, 445, 4E2, 517, 547, 673, 713, 809, 964, X34, X63, X85, E24, E59, ... |
105 | 5, 7, 58, 67, 8X, 198, 227, 35X, 39X, 54E, 5E3, 74E, 975, 9X7, 9E6, E0E, E88, E91, EE7, ... | 14, 33, 43, 89, 123, 134, 165, 178, 1X9, 242, 26X, 278, 446, 548, 653, 6X6, 880, 8EX, 9XE, 9E3, X27, E38, ... |
107 | 25, 47, 49, 114, 128, 166, 193, 264, 2E7, 307, 456, 470, 4X1, 5EE, 74X, 798, 7E8, 847, 911, 959, 972, X34, X3X, ... | 103, 146, 23X, 268, 26E, 370, 507, 78X, 854, 924, 954, 9X1, X72, X99, XXX, E97, ... |
111 | 48, 5E, 64, 131, 13X, 225, 242, 2X5, 2E6, 429, 49X, 562, 565, 680, 700, 723, 753, 811, 8E1, 8EX, 905, 90X, 990, X37, X91, XX0, ... | 17, 29, 39, E0, E1, 111, 15X, 1E5, 353, 487, 525, 549, 560, 6X9, 845, 988, X41, E3E, E75, ... |
117 | 26, 52, 9X, E7, 103, 203, 320, 52E, 596, 620, 632, 650, 721, 766, E68, E98, ... | 17, 44, X6, 104, 145, 188, 196, 272, 282, 392, 39E, 458, 494, 515, 571, 612, 654, 7X6, 7E5, 804, 871, 986, 9XX, X44, X64, EE4, ... |
11E | 38, 39, X7, 127, 132, 297, 315, 319, 338, 3E7, 406, 4X4, 525, 538, 655, 679, 734, 88E, 9E5, XX3, E13, E16, ... | 2, 32, 52, 71, 12X, 197, 1E2, 260, 2XE, 407, 513, 5E8, 617, 651, 65X, 755, 842, 863, X9X, E07, EX1, ... |
125 | 50, 52, E7, E9, 213, 221, 268, 2E5, 392, 473, 7E7, 857, 993, X11, E97, ... | 14, 26, 1E5, 232, 286, 49E, 510, 57E, 936, E77, EE2, ... |
12E | 214, 33X, 40X, 666, 674, 693, 7XX, ... | E, 47, 193, 201, 229, 32E, 605, 866, 9E7, X07, E03, ... |
131 | 5, 31, 123, 2E7, 365, 3E7, 436, 475, 47E, 556, 718, 82E, X86, E79, ... | 201, 388, 408, 448, 468, 47X, 704, 715, 72X, 7X5, 8E4, X19, E19, E57, EE4, ... |
13E | 62, 15X, 2X8, 339, 417, 474, 4X3, 4E4, 51E, 650, 7E8, 872, ... | 2, 30, 64, 37E, 3EE, 422, 7X8, 846, X40, X47, E50, ... |
141 | 9X, 211, 346, 3X2, 451, 481, 514, 756, 83X, XX8, ... | 10, 2X0, 353, 426, 469, 644, 705, 791, 823, 871, X74, X9X, ... |
145 | 29, 178, 188, 19X, 23E, 263, 284, 415, 537, 579, 726, 739, 944, 952, 9XX, X87, ... | 44, 46, 140, 283, 309, 453, 57X, 873, 946, X01, X62, X86, E26, E76, E7X, ... |
147 | 110, 262, 27E, 295, 30X, 446, 459, 495, 518, 649, 660, 668, 69E, 6X4, 794, 831, 848, E4X, ... | 2, E4, 153, 217, 234, 2E4, 32E, 349, 479, 525, 574, 592, 774, 887, 892, 939, E1E, E3E, E4X, ... |
157 | 3X, 49, 256, 33X, 38E, 405, 465, 591, 597, 695, 764, 785, 7E8, 91X, 967, E91, ... | 56, 59, 202, 456, 471, 538, 5X1, 67E, 69X, 819, 935, X14, X63, E06, E80, E9E, ... |
167 | 133, 136, 163, 165, 471, 525, 635, 642, 717, 956, 988, E32, ... | 9, 3X, 44, 90, 177, 261, 364, 413, 4X3, 514, 522, 551, 63X, 650, 654, 706, 75X, 840, 941, X56, X92, ... |
16E | 60, E4, 177, 180, 18E, 22X, 252, 358, 377, 3X4, 401, 47E, 494, 51E, 687, 793, 7XX, 847, 856, 938, X07, X67, X93, E51, ... | 1X, 22, 49, 68, X9, 12X, 186, 295, 35X, 594, 652, 701, 703, 80X, 905, 915, 921, 944, 948, 94X, 9X3, ... |
171 | 426, 505, 52X, 5E6, 672, XEX, E19, ... | 5, E, 47, 85, 14X, 310, 334, 5X3, 704, 708, 749, 7X1, X93, E04, E36, ... |
175 | 422, 982, 985, ... | 349, 556, 770, 813, X70, E79, ... |
17E | 167, 198, 267, 336, 3E0, 5EX, 9E3, ... | 59, X6, 159, 34X, 499, 5X5, X57, XE7, E5E, ... |
181 | 97, 117, 167, 1X1, 1X6, 236, 495, 5X9, 706, 71X, 9E3, XX4, ... | E5, 162, 24X, 43X, 487, 49X, 540, 61X, 670, 70X, 760, E48, EE6, ... |
18E | 31, 186, 1X3, 436, 659, 796, ... | 14, 37, 163, 218, 30E, 3E3, 460, 523, 577, ... |
195 | 44, 66, 303, 323, 46X, 4E1, 712, 717, 839, 854, 886, 8X3, 96X, X42, E64, E80, ... | 30, 123, 197, 268, 344, 347, 484, 740, 83E, 881, 907, E09, E45, ... |
19E | 88, XE, 115, 338, 352, 3XE, 513, 5X2, 639, 6X3, 76E, 794, X84, ... | 80, X0, 10X, 193, 1X3, 217, 294, 319, 449, 471, 560, 66X, 815, 929, 971, X04, E2E, ... |
1X5 | 35, 40, 206, 351, 374, 578, 5X3, 80E, ... | 64, 128, 397, 43E, 48X, 530, 5X2, 718, 734, 767, 7X2, 7X4, 83X, 879, X41, E49, E5E, EE2, ... |
1X7 | 6, 19, 136, 149, 166, 180, 40X, 43X, 440, 768, 845, 880, 885, 989, X25, X53, ... | 99, 2X3, 312, 446, 572, 580, 635, 6E5, 964, EX5, ... |
1E1 | 242, 335, 451, 664, 665, 696, E19, ... | 22, 37, 60, 148, 15E, 276, 2XX, 413, 5X0, 621, 677, 688, E97, ... |
1E5 | 161, 312, 426, 436, 55X, 600, X98, EE2, ... | 3, 101, 399, 562, 709, 8E0, ... |
1E7 | 11, 145, 192, 200, 22E, 272, 298, 667, 784, 957, X45, ... | 113, 124, 174, 22X, 22E, 26X, 283, 327, 45E, 507, 602, 674, 693, 6X7, 73X, 773, X20, ... |
205 | E4, 284, 333, 8XE, ... | X, 6X, 84, 97, X2, 345, 62E, 7X7, X15, ... |
217 | 164, 177, 224, 256, 27E, 325, E98, ... | 14, 7X, 255, 342, 36E, 645, 825, X96, ... |
21E | 116, 404, 695, 845, XX5, XX9, ... | 155, 246, 30E, 351, 547, 598, 651, 726, 901, X86, X93, E18, ... |
221 | 2E, 87, 1X7, 327, 435, 924, ... | 2, 52, 147, 172, 20X, 376, 634, 694, ... |
225 | X, 10, 105, 141, 1XE, 288, 394, 3X6, 435, 4E9, 60X, 725, 767, X6E, EE1, ... | 14, 65, X7, 100, 104, 189, 31X, 478, 648, 702, 773, 837, 99E, E72, ... |
237 | 162, 187, 353, 387, 67X, 790, 906, 940, 94E, X0E, X71, X82, ... | 1E, 3X, 115, 212, 245, 367, 370, 587, 594, 649, E23, E65, ... |
241 | 17, 85, 93, 24E, 299, 348, 3E8, 420, 889, 949, X10, XE4, ... | 1X9, 211, 268, 741, 968, ... |
24E | 22, 42, 118, 12X, 2XE, 476, 507, 520, 60E, 634, 888, 8X8, 92X, 94X, X1X, E3X, E95, ... | 2, 5, 19, 22, 203, 40X, 5X9, 9X1, X31, X84, ... |
251 | 33, 40, 404, 434, 462, 519, X7E, E47, ... | 324, 454, 473, 646, 657, 69X, 794, 801, 887, E54, ... |
255 | 10, E2, 11X, 149, 453, 512, 580, 997, ... | 1X, E5, 17X, 588, 684, 6X7, 85E, 892, 92X, 97E, ... |
25E | 1X, 150, 2E6, 392, 3X0, 789, ... | 3, 51, 70, 262, 34E, 412, 480, 746, 772, 949, ... |
267 | 57, 52E, 621, 738, 7E9, 876, 915, 929, X00, X87, E50, EX4, ... | 30, 107, 1XX, 439, 600, 628, 677, 678, 9E2, X02, ... |
271 | X0, 11E, 84E, 880, 929, X20, X40, ... | 24, 278, 3X1, 551, 737, 741, 832, X44, ... |
277 | 143, 3XX, 4X1, 753, 7E7, 868, ... | 235, 236, 288, 2E0, 3E9, 400, 459, 5XX, 6E2, 90X, 923, EX2, ... |
27E | 40, 83, 118, 20E, 250, 25E, 297, 337, 472, 524, 565, 5X7, 609, 664, 869, 882, 8X5, X40, E05, E84, ... | 37, 93, 119, 169, 191, 232, 2X2, 385, 4E7, 613, 633, 777, 7X7, 883, X21, X7E, X98, E67, EE8, ... |
285 | 46, 89, 2X2, 2EX, 317, 4E7, 546, 559, 781, 983, 9E5, X41, EE1, ... | 59, 392, 445, 521, 5X0, 99X, 9E4, ... |
291 | 327, 443, 4X9, 781, 817, ... | 72, 77, 198, 205, 312, 469, 916, ... |
295 | 32, 168, 24E, 342, 34X, 728, 736, 952, 9X2, XX3, E32, ... | 1X7, 345, 647, 709, 727, 822, 894, 952, 9E0, E28, ... |
2X1 | 35, 351, 46E, 735, ... | 290, 2E9, 35E, 754, 831, 911, XE7, E29, ... |
2XE | 15, 258, 66X, ... | 24, 129, 1XX, 3E8, 540, 581, ... |
2E1 | 574, 597, 5X2, 9EX, ... | 6E, 1X8, 235, 623, 786, 8X6, 914, 9X7, E50, EE1, ... |
2EE | 298, 534, 6E8, 926, 949, ... | 212, 248, 47X, 555, 599, 817, 87E, EX2, ... |
301 | 3X, 2E5, 3X3, 5X5, 609, 657, 786, E09, E48, E95, ... | 155, 202, 406, 453, 455, 522, 553, 711, 780, X80, ... |
307 | 64, 78, 165, 502, 529, 922, X88, E28, ... | E, 68, 14E, 1EX, 285, 666, 75X, 940, 954, X87, XX5, E14, ... |
30E | 561, 768, ... | 210, 651, 6E6, ... |
315 | 32, 90, 10E, 155, 167, 483, 531, ... | 113, 131, 270, 31X, 562, 65X, 801, 840, 881, ... |
321 | 24, 29E, 2X6, 501, X6E, E73, ... | 67, 250, 307, 473, 484, 49E, 822, E59, ... |
325 | 15E, 170, 18E, 2X6, 459, 463, 501, 525, 6X1, 914, X11, ... | 27, 84, 3E9, 400, 4E0, 4E8, 613, 718, 751, 7EX, 87E, 97E, 9E5, E6X, ... |
327 | 31, 240, 464, 594, 801, 811, 836, 94X, EX3, ... | 237, 36X, 750, 906, X71, ... |
32E | 178, 483, 553, 5XX, 614, 682, 75E, E7E, ... | 44, 73, XX, 456, 521, 91X, ... |
33E | 4E, 214, 296, 358, 515, 631, 7E9, 850, E72, EX9, ... | 128, 13E, 499, X47, X97, E6E, ... |
347 | 13, 195, 1EX, 2X8, 438, 4E7, 592, 63E, 67X, ... | 3, 350, 73X, E2E, E52, ... |
34E | 36X, 917, ... | 24, 67, 7E8, 9XE, ... |
357 | 198, 4X7, 992, ... | 161, 215, 222, 536, 695, ... |
35E | 356, 389, 397, 424, 4E9, 588, 62E, 687, ... | 12, 2E, 37, 59, 612, 873, 8E0, X38, X65, ... |
365 | 6, 72, 132, 600, 836, X95, E08, ... | 2X2, 846, 905, ... |
375 | 2, 31, 325, 402, 436, 5X2, 6E2, 739, 7X3, ... | 190, 3E3, 47X, 48X, 6X6, 6E6, 922, 968, X05, E19, ... |
377 | 7E, 572, 676, ... | 4EX, 650, 6XX, 9EE, X1X, ... |
391 | 3, 34, 5X, 182, 190, 311, 3X7, 457, 463, 535, 5E0, 9E4, X33, X84, E0E, ... | 118, 223, 268, 476, 581, 601, 67X, 75E, 835, 864, E10, E50, ... |
397 | 335, 37E, 480, 785, ... | 9, 26, 6E, 185, 3X9, 594, 784, 9EE, X78, E39, E40, ... |
3X5 | 2X9, 6X0, 824, 921, 944, ... | E, 19X, ... |
3XE | X3, 192, 42E, 544, 632, 801, 8E1, 984, ... | 300, 347, 750, 776, 884, X1E, X83, E16, E64, EEX, ... |
3E5 | 26, 96, 472, 536, 568, 832, 854, XX8, ... | 125, 192, 276, 661, X34, ... |
3E7 | 75, 103, 243, 3X9, 655, 771, 821, ... | 79, 242, 864, 955, 971, 9X7, E83, ... |
401 | 74, E8, 113, 147, 979, X44, X47, E95, ... | 3, E6, 263, ... |
40E | 178, 476, 59X, 5X3, 625, 746, 781, EX5, EX6, EXX, ... | 1E, 140, 149, 381, 482, 672, 72E, ... |
415 | 64, 23E, 566, X77, X91, ... | 149, 32X, 662, ... |
41E | X4, E1, 279, 301, 382, 659, 8E6, X91, ... | 128, 357, 404, 433, 813, ... |
421 | ... | 79, 10X, 17E, 375, X14, E79, ... |
427 | 2, 571, 691, 757, 982, E57, E90, ... | 439, 892, 89E, ... |
431 | 140, 155, 249, 28E, 455, 458, 61X, 922, ... | 7X, 184, 197, 210, 22E, 473, 672, ... |
435 | 137, 215, 44E, 482, 982, X76, ... | 2E, 1X0, 317, 626, 804, X19, ... |
437 | 5, 16X, 22E, 314, 5E1, ... | 10X, 1X7, 555, 762, X46, X77, ... |
447 | 33, 55, 475, X03, E7X, ... | 17, 339, 624, 830, X07, E75, ... |
455 | 897, ... | X, ... |
457 | 13X, 452, 672, 709, 821, X25, ... | 56, 340, 355, 4E0, 529, 90E, 984, ... |
45E | 229, 65E, 809, E23, ... | 124, 35E, 5X4, 614, 838, 839, ... |
465 | 20, ... | 56E, 86E, ... |
46E | 67, 244, 653, ... | X59, E64, ... |
471 | 20, 42, 49, 510, 565, 7E8, 936, 972, 9X8, X68, X7X, ... | 66, 117, 732, ... |
481 | 86, 13X, 2E5, 5E7, 905, ... | 88, 351, 352, 620, 993, E26, ... |
485 | 85, 544, 722, 91X, 969, X90, ... | 256, 2E9, 405, 659, 770, 969, X30, X34, ... |
48E | 358, 359, ... | 201, 33X, 381, 677, 67X, 71E, 908, X9E, ... |
497 | 92, 677, 784, E64, E72, ... | 35, 502, 786, ... |
4X5 | 10, 94, 162, 193, 477, 57X, 6X1, 755, X44, E89, ... | 2, 69, 233, 495, 535, 537, 599, 996, 9E7, ... |
4E1 | 96, 1X7, 399, 585, 897, E26, ... | 18, 2E, 36, 74, 163, 781, 8E2, EE7, ... |
4EE | 1E7, 27X, ... | 2EX, 6E3, X26, ... |
507 | 6E8, X70, ... | 140, 1X7, 31X, 458, X06, ... |
511 | 3E2, 888, 976, E86, ... | 16, 402, ... |
517 | 63, 3X3, 653, E55, E99, ... | X3, 289, 2X4, 3XX, ... |
51E | 292, 387, ... | 684, ... |
527 | 34, 15E, 397, 6E8, 737, 760, ... | 248, 2E5, 474, 813, ... |
531 | 52, 511, 893, X17, ... | 58, 393, ... |
535 | 5X, 522, 894, ... | X44, EE4, ... |
541 | 51, 322, 3X4, 477, 783, X43, ... | 105, 112, 321, 456, 638, 809, 98E, ... |
545 | 8X4, ... | 9, 58, 374, 411, 454, 888, X39, ... |
557 | 268, 286, 422, X41, ... | 307, 354, 70E, 774, 90E, X38, XX2, XX9, ... |
565 | 339, 949, X85, ... | 18, 9E, 345, 446, 597, 8XX, 986, ... |
575 | 582, E6X, ... | 10X, 111, 128, ... |
577 | 246, 311, 410, 562, 933, X13, XE4, ... | 6, 91, 272, 432, 539, XE3, ... |
585 | 161, 4X4, 580, 64E, 6E1, 7XX, ... | 28E, 426, 461, 580, 831, 85X, 90E, X43, ... |
587 | 120, 909, 9EX, ... | 27E, 459, 574, ... |
58E | 9E, 4EE, 504, 8E4, 9E3, ... | 12E, 178, 447, 57E, 965, ... |
591 | 14X, 1X7, 529, 73X, ... | 26, 69, 305, 337, 581, ... |
59E | 47, 210, 3X6, 3E3, 430, 530, X30, ... | 88, 273, 44X, 64X, 880, ... |
5E1 | 2EX, ... | 181, 318, ... |
5E5 | 1X, 102, 858, 955, ... | 22, 2E, 167, 18X, 1X6, 965, E52, ... |
5E7 | 306, 6X3, 792, ... | 372, 589, 679, 893, 8E0, X5X, X86, E72, ... |
5EE | X83, ... | 204, 22E, 490, 896, 973, 9EX, ... |
611 | 601, 682, ... | 369, ... |
615 | 1XE, 49E, 592, 5X2, 897, E94, ... | 267, 412, 985, XX7, ... |
617 | 11, 192, 398, 990, X99, ... | 155, 669, X37, ... |
61E | 244, 3E2, 427, 684, 870, XX5, ... | 310, 46X, 473, 482, 966, X23, XX5, ... |
637 | E, 338, E04, ... | ... |
63E | 12X, 520, 575, 669, ... | 367, XEX, ... |
647 | 638, 880, ... | 11, 657, 662, 744, 974, ... |
655 | 5, 58E, 653, ... | 374, 927, ... |
661 | 590, 609, 634, ... | 50, 146, 402, 703, ... |
665 | 180, 2X9, 52X, ... | 16, 66E, E61, ... |
66E | 190, 471, 889, 8E4, E39, ... | 28X, 430, ... |
675 | 174, ... | ... |
687 | 195, 2E7, 527, 993, ... | 15, 196, 527, 656, 710, 896, 928, ... |
68E | 162, 594, X20, ... | 70, X74, X79, ... |
695 | E35, E37, ... | 921, ... |
69E | 103, 2E2, ... | 8E, 227, 355, 362, 462, 9X7, E56, ... |
6X7 | 11, E1, 2X7, 625, 904, ... | 46, 4E, 957, ... |
6E1 | 359, 3EE, 413, 466, 623, 624, X48, ... | 167, 737, 9X8, ... |
701 | E5, 236, 492, 839, 849, 9X6, ... | 9, 74X, XX5, ... |
705 | 5X, E79, ... | ... |
70E | 1E4, 204, 897, E12, ... | ... |
711 | 11, 531, 75E, 991, 9E1, X07, E90, ... | 737, E14, ... |
71E | X, ... | 24, 69, 88, 856, ... |
721 | 798, 870, ... | 69, 972, 978, E00, ... |
727 | 231, 320, 404, 610, 836, ... | 6X6, ... |
735 | 6, 836, 83E, 924, ... | 144, 191, 212, 500, 580, E56, ... |
737 | 867, E17, ... | 47X, 494, E12, E7X, ... |
745 | 17, 16E, 2X3, 685, 749, 8E8, E85, ... | 27, X7, 18X, 379, 52E, ... |
747 | X58, X59, E52, ... | 32, 48, 109, 210, 288, ... |
751 | 72, 55X, 731, 99X, ... | 152, 81E, ... |
767 | 116, 5EX, 729, ... | 70, 143, 364, 536, 60X, ... |
76E | 3, 276, 4X7, ... | 13, 444, 44E, ... |
771 | 17X, 27E, 380, ... | 57E, 664, E67, ... |
775 | 474, 487, ... | 16, 163, 210, 448, 663, X84, ... |
77E | 240, 314, ... | 75, ... |
785 | 1X3, 257, 527, 701, 716, 852, ... | 118, 42E, 516, 95X, ... |
791 | 68X, 881, 973, ... | 171, 630, E6E, ... |
797 | X71, ... | 2E5, 536, ... |
7X1 | ... | 47, 2X2, 805, 910, 955, X58, E6E, E84, ... |
7EE | 240, ... | 11, 4XE, 692, 730, 9XE, ... |
801 | 286, 6E3, 976, ... | 42, 13X, 637, 967, 9XE, ... |
80E | 288, ... | 18, 13E, 292, 397, 579, 673, 784, 865, 8E6, X80, ... |
817 | E3, 29X, 36X, X06, X42, ... | E2, 669, ... |
825 | 85, E1, 732, 891, X39, ... | 7X5, ... |
82E | 411, 68E, ... | 89, 2X9, 547, X05, ... |
835 | 11, 91, 221, 226, 55X, 6X0, X27, ... | 254, 36X, 8X8, ... |
841 | 104, 1E6, 217, 34E, 61E, ... | 257, ... |
851 | 11E, 1X7, 558, 876, E70, ... | X5E, ... |
855 | 31X, 429, 459, ... | 34, 10X, 173, 256, 40X, 45E, 8X5, ... |
85E | 250, 274, 5X6, ... | 46X, 5EE, ... |
865 | 154, 440, ... | 12, 731, 809, ... |
867 | EX, 639, 6E6, ... | 253, 49X, 538, 61E, 8E8, ... |
871 | 122, 2X8, 805, X90, ... | 235, 240, 98X, E7E, ... |
881 | 189, 442, 569, ... | 21, ... |
88E | 865, ... | 326, 63X, 653, ... |
8X5 | 34, 162, ... | 169, 45X, ... |
8X7 | 2, 727, ... | 6X, 26X, 4EX, X32, ... |
8XE | 470, 540, 671, X7E, E03, EX2, ... | 417, 484, ... |
8E5 | 11X, 786, 91E, ... | ... |
8E7 | 2X4, ... | X16, ... |
901 | 2E, 561, ... | 680, 9X4, ... |
905 | 806, ... | 110, 11E, 650, 94X, 95E, ... |
907 | 40, 672, 992, ... | E0, 208, 342, 373, 556, 960, X80, ... |
90E | 163, 4X1, ... | 128, 570, 5E7, 877, X74, ... |
91E | 36, 2X8, 38X, X50, ... | 206, 654, ... |
921 | 2E4, 6X5, 801, ... | 47, 450, 677, 824, ... |
927 | 5EE, ... | 456, 615, 7E0, ... |
955 | 4E9, 803, 897, E9X, ... | 7XX, ... |
95E | 3, 452, 487, 650, ... | 257, 799, XX0, ... |
965 | 384, 453, 512, ... | 74, 18X, 205, 4X4, ... |
971 | 176, 17X, 4E7, 8E3, ... | ... |
987 | 13E, 65E, E75, E85, ... | 1E2, 43X, 893, 982, XE1, XEX, E11, ... |
995 | 97, 83E, ... | ... |
9X7 | 24, E3X, ... | 240, 251, E43, E76, ... |
9XE | ... | 510, ... |
9E1 | 572, 5X5, ... | 415, 71X, 738, ... |
9E5 | 113, 196, ... | 362, 385, 78X, ... |
9EE | 293, 865, ... | 102, 4X2, 680, X3E, X7E, ... |
X07 | 5X9, ... | 4X, 2E4, 488, 5E5, 878, 9X6, ... |
X0E | ... | 924, ... |
X11 | 297, ... | 560, 74E, 878, ... |
X17 | 987, ... | 16X, 259, 491, 605, ... |
X27 | 3X6, 916, ... | E9, 321, 623, X91, ... |
X35 | ... | 41, 98, E78, ... |
X37 | 312, 747, ... | 10, 115, 30E, 671, 71X, ... |
X3E | 18, 57, 114, EXX, ... | 132, 543, 937, ... |
X41 | 423, X43, ... | 296, 95X, E81, ... |
X45 | 2X, ... | ... |
X4E | 25X, 356, 53E, 855, ... | 2E, X1, ... |
X5E | 91E, ... | XE9, ... |
X6E | 316, 4EE, ... | 946, ... |
X77 | 4E7, 579, 691, ... | 1EX, 687, ... |
X87 | 6X4, 8XE, X8X, ... | 26E, 719, E30, ... |
X91 | 87, 228, 404, XX3, ... | ... |
X95 | ... | 538, 596, 9X3, ... |
X9E | 996, ... | ... |
XX7 | 843, ... | 643, ... |
XXE | 45, 228, 32E, ... | 241, 32E, 398, ... |
XE7 | 725, 957, XX3, E13, ... | 3, 89X, ... |
XEE | 5E, 154, 42X, 57E, E52, ... | ... |
E11 | 465, 668, 7X6, ... | 32, 483, ... |
E15 | 852, ... | ... |
E1E | ... | 351, 5E8, ... |
E21 | 132, 340, 396, 3X0, 558, 732, ... | 238, 413, 486, 914, ... |
E25 | 264, 433, 63X, 80X, E47, ... | 3E1, ... |
E2E | 328, ... | 7, ... |
E31 | 1E1, 275, 515, ... | 5X8, 692, 6XX, E50, ... |
E37 | 3, 603, X45, ... | 89, 3X3, 91E, XE6, ... |
E45 | 117, 54E, ... | 36, 363, 425, 561, ... |
E61 | ... | 166, E1X, ... |
E67 | 222, 453, 950, EE6, ... | 3, 2X0, 478, 674, 782, ... |
E6E | 33X, 90E, ... | 50, 687, 697, ... |
E71 | 3X4, X92, ... | 16X, X10, ... |
E91 | 81, X0, X3, ... | 150, 962, X37, ... |
E95 | 94, 45X, 988, ... | 332, 576, ... |
E97 | 7, 127, 268, 2XE, 404, 527, 899, X4X, E73, ... | 289, 618, 779, 955, E52, ... |
EX5 | 2E8, 40E, 424, 575, 621, ... | 2, 21, 542, X85, ... |
EE5 | 92, E8, 222, 521, 834, ... | 58X, ... |
EE7 | 375, 629, ... | 8XE, 970, ... |
The generalized repunit conjecture
A conjecture related to the generalized repunit primes:[1][2] (the conjecture predicts where is the next generalized repunit prime, if the conjecture is true, then there are infinitely many repunit primes for all bases ) For any integer , which satisfies the conditions:
- .
- is not a perfect power. (since when is a perfect th power, it can be shown that there is at most one value such that is prime, and this value is itself or a root of )
- is not in the form . (if so, then the number has aurifeuillean factorization)
has generalized repunit primes of the form
for prime , the prime numbers will be distributed near the best fit line
- ,
where limit , and there are about
base repunit primes less than .
- is the base of natural logarithm.
- is Euler–Mascheroni constant.
- is the logarithm in base
- is theth generalized repunit prime in base (with prime)
- is a data fit constant which varies with.
- if, if.
- is the largest natural number such that is ath power.
We also have the following 3 properties:
- The number of prime numbers of the form (with prime ) less than or equal to is about .
- The expected number of prime numbers of the form with prime between and is about .
- The probability that number of the form is prime (for prime ) is about .
Demlo numbers
Kaprekar has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit[3]. They are named after Demlo railway station 30 miles from Bombay on the then G.I.P. Railway, where Kaprekar started investigating them. He calls Wonderful Demlo numbers those of the form 1, 121, 12321, 1234321, ..., 123456789XEX987654321. The fact that these are the squares of the repunits has led some authors to call Demlo numbers the infinite sequence of these[4], 1, 121, 12321, 1234321, ..., 123456789XEX987654321, 123456789E00EX987654321, 123456789E0120EX987654321, ..., although one can check these are not Demlo numbers for n = 10, 1E, 2X, ...
- ↑ Deriving the Wagstaff Mersenne Conjecture
- ↑ Generalized Repunit Conjecture
- ↑ Template:Harvnb, Template:Harvnb
- ↑ Template:MathWorld