FANDOM


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.

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 Edit

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):

$ R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b), $

where

$ \Phi_d(x) $ is the

$ d^\mathrm{th} $ cyclotomic polynomial and d ranges over the divisors of n. For p prime,

$ \Phi_p(x)=\sum_{i=0}^{p-1}x^i, $

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

$ \Phi_2(x) $ and

$ \Phi_3(x) $ and

$ \Phi_4(x) $ and

$ \Phi_6(x) $ and

$ \Phi_{10}(x) $ are

$ x+1 $ and

$ x^2+x+1 $ and

$ x^2+1 $ and

$ x^2-x+1 $ and

$ x^4-x^2+1 $

, 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 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 basen 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 Edit

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 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.
  • 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 Edit

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 Edit

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 Edit

(Prime factors colored red means "new prime factors", i.e. the prime factors dividing Rn but not dividing Rk for any k < n)

nfactorization of Rnnumber of prime factors of Rn (counted with multiplicity)number of distinct prime factors of Rnnumber of "new prime factors" (i.e. the prime factors dividing Rn but not dividing Rk for any k < n) of Rn
11000
211111
3111111
45 × 11 × 25332
511111111
67 × 11 × 17 × 111442
746E × 2X3E222
85 × 11 × 25 × 75 × 175552
931 × 111 × 3X891332
X11 × E0E1 × 11111331
EE × 1E × 754E2E41333
105 × 7 × 11 × 17 × 25 × 111 × EE01771
111E0411 × 69X3901222
1211 × 157 × 46E × 2X3E × 7687552
1351 × 111 × 471 × 57E1 × 11111553
145 × 11 × 15 × 25 × 75 × 81 × 175 × 106X95883
15X9X9XE × 126180EE0EE222
167 × 11 × 17 × 31 × 111 × E61 × 1061 × 3X891882
171111111111111111111111
1852 × 11 × 25 × E0E1 × 11111 × 24727225761
19111 × 46E × 2X3E × E00E00EE0EE1441
1XE × 11 × 1E × 754E2E41 × E0E0E0E0E1551
1E3E × 78935EX441 × 523074X3XXE333
205 × 7 × 11 × 17 × 25 × 75 × 111 × 141 × 175 × EE01 × 8E5281EE2
2111111 × 1277EE × 9X06176590543EE332
22112 × 67 × 18X31 × X8837 × 1E0411 × 69X3901763
2331 × 111 × 3X891 × 129691 × 9894576430231552
245 × 11 × 25 × 157 × 46E × 481 × 2X3E × 7687 × 2672288X41992
254E × 123EE × 15960E × 160605E10497012E4E444
267 × 11 × 17 × 27 × 51 × 111 × 2E1 × 471 × 57E1 × E0E1 × 11111 × 1878710103
27271 × 365E0031 × 464069563E × 39478E3664E444
285 × 11 × 15 × 25 × 75 × 81 × 175 × 75115 × 106X95 × 1748E3674115XX2
29E × 1E × 111 × 368E51 × 2013881 × 754E2E41 × 16555E1X1773
2X11 × 1587 × X9X9XE × 126180EE0EE × 7605857409257552
2E5E × 34E × 46E × 2X3E × 11111 × 32XXE1 × 205812E × EX59849E885
305 × 7 × 11 × 17 × 25 × 31 × 61 × 111 × E61 × 1061 × EE01 × 3X891 × 1E807X62E6111112
311398641 × 9E2X6732EE74552406X78E76247691222
3211 × 1XE7 × 4901 × 127543624027 × 1111111111111111111553
33111 × 19491 × 1E0411 × 5XE48X1 × 69X3901 × 1064119E745041663
3452 × 11 × 25 × 35 × 75 × 175 × 375 × E0E1 × 11111 × 62041 × 1X7X9741 × 2472722511104
356E × 472488E21 × 4E2EX47X7863X18E5E18253377315E333
3672 × 11 × 17 × 37 × 111 × 157 × 46E × 2X3E × 7687 × 9X17 × 76E077 × E00E00EE0EE111103
372EE × 4159911 × 273263674E × 4X748X0X65EXX3943375X351444
385 × E × 11 × 1E × 25 × 1461 × 2181 × 3801 × 754E2E41 × E0E0E0E0E1 × 113006390X1EE4
3931 × 51 × 111 × 471 × 57E1 × 11111 × 15991 × 3X891 × 1905201 × 7229231 × 7843701EE4
3X11 × 3E × 591 × 7231 × 78935EX441 × 523074X3XXE × 3266712021E531E1773
3E832966217X8X111 × 16EE6202E02X5311278504010EX13001222
405 × 7 × 11 × 15 × 17 × 25 × 75 × 81 × 111 × 141 × 175 × 4541 × EE01 × 1E601 × 106X95 × 8E5281 × 14660948115153
4146E × 2X3E × 38E01 × 3257955345X23186304E321167X366X2593101442
4211 × 1167 × E0E1 × 11111 × 2E0X1 × 36967 × 1277EE × 102X155X1 × 9X06176590543EE994
43111 × X9X9XE × 6E50611 × 126180EE0EE × 16EEX75X0381X76766164247X1552
445 × 112 × 25 × 45 × 67 × 485 × 18X31 × X8837 × 1E0411 × 69X3901 × 6X78XX0721X7626395X110E3
458E × 51E × 698X51 × 9X6X571 × 41X866326E31 × 191017735473010X59437791666
467 × 11 × 17 × 31 × 91 × 111 × 1X7 × 347 × E61 × 1061 × 1X761 × 3X891 × 129691 × 13X29831 × 989457643023113135
47E × 1E × 3081 × 11111 × 754E2E41 × 417569335X9871 × X54106288E1178834252681773
485 × 11 × 25 × 75 × 157 × 175 × 46E × 481 × 2X3E × 7687 × 2672288X41 × EEEE0000EEEE0000EEEE000110101
49111 × 55E81 × 630E1 × 2376268253771 × 180218815260491 × 1111111111111111111664
4X11 × 4E × 123EE × 15960E × 331887791 × 160605E10497012E4E × 348EX480E981E13E8E21772
4E1092E × 13167124X1E5E6E24E1 × 991027128X344E58009X13377728X377816E333
5052 × 7 × 11 × 17 × 25 × 27 × 51 × 111 × 2E1 × 471 × 57E1 × E0E1 × EE01 × 11111 × 18787 × 24727225 × 100EEEXEXEE00010116151
516X3531 × 23E214X01 × 9X134140XE002419X65090184E86425024X210X79953X1333
5211 × 271 × 50E237 × 365E0031 × 464069563E × 39478E3664E × 2221710X303X15671780413X7772
5331 × 111 × 46E × 2X3E × 3X891 × E00E00EE0EE1 × EEE000000EEE000000EEEEEE000EEEEEE001771
545 × 11 × 15 × 25 × 75 × 81 × 175 × 541 × 75115 × 106X95 × 1748E3674115 × 22E6E3E4614X2X0E739X6493730X8110102
55XE × 11111 × 1E0411 × 69X3901 × 45E152651 × 328X222960EE4296E × X003996X2X7736694E501774
567 × E × 11 × 17 × 1E × 57 × 111 × 147 × X11 × 3207 × 368E51 × 2013881 × 754E2E41 × 16555E1X1 × 76E4545077 × E0E0E0E0E114145
5713286641 × 3176XE592E × 118357X16417E44E × 2X7304X3E254X17927452584836E1769951444
585 × 11 × 25 × 1587 × X9X9XE × 126180EE0EE × 1349X9E47X1 × 7605857409257 × 93503726E44887X0575721992
593E × 111 × 78935EX441 × 523074X3XXE × 29E724E34313E08941 × 3X7E21516142789517140EE6X71662
5X11 × 5E × 157 × 34E × 46E × 2X3E × 7687 × E0E1 × 11111 × 32XXE1 × 205812E × EX59849E × 10EEEXXXE011110EXXXE0001111111
5E2E61 × E411 × 48214E17105E9X31322076E52260904296018248289388E3E03E1E49225X961333
605 × 7 × 11 × 17 × 25 × 31 × 61 × 75 × 111 × 141 × 175 × E61 × 1061 × EE01 × 3X891 × 8E5281 × 8E7701 × 86X069E01 × 1X59306601 × 1E807X62E6118183
6133553026E085851E × 3E944878XE2950EX202931666X2779X976062X268389208736281XE8E222
6211 × 1398641 × 19122X7 × 5705544E4727 × 11648541252X9133271 × 9E2X6732EE74552406X78E76247691663
6351 × 111 × 421 × 471 × 57E1 × 11111 × 1277EE × 22XX6541 × 9X06176590543EE × 13479X713133E72X42X33465X235661XX3
645 × 11 × 25 × 1XE7 × 4901 × 7E45X1 × 127543624027 × 17962521E22801 × XXXX48E48508X7121 × 1111111111111111111XX3
65E × 1E × 46E × 2811 × 2X3E × 754E2E41 × X6E63531 × 3108049E09E × 33473888220964X1 × 5635793E280446201E1084EXX5
667 × 112 × 17 × 67 × 111 × 221 × 5E7 × 18X31 × 19491 × X8837 × 1E0411 × 5XE48X1 × 69X3901 × 3547080331 × 35X1236E57 × 1064119E74504115144
6779991 × 3784511E × 562747X052E4X17641X27X25E9EX5E563E27602431946E43634793X292763X675E333
6852 × 11 × 15 × 25 × 35 × 75 × 81 × 175 × 375 × E0E1 × 11111 × 62041 × 106X95 × 1X7X9741 × 24727225 × EEEEEEEE00000000EEEEEEEE0000000115141
6931 × 111 × 3X891 × 129691 × 1EX4391 × 9894576430231 × 6206698749E301 × E89X7X727394X9E4477E48607435213231883
6X11 × 6E × 7151 × 3E022X7X7 × 472488E21 × 1646X9EEE9941 × 31404E9E46699E07 × 4E2EX47X7863X18E5E18253377315E884
6E11E × 6E1 × 11X1 × 1093E × 136X04861 × 69E54141E1 × 38148576964E × 79E54E87686014E × 8EE2415483165180830X595031999
705 × 72 × 11 × 17 × 25 × 37 × 111 × 157 × 46E × 481 × 2X3E × 7687 × 9X17 × EE01 × 76E077 × 2672288X41 × E00E00EE0EE1 × 100EEEXEE0000EEEXEE00010117161
71711 × 10X7E × 11111 × X9X9XE × 17508X1 × 126180EE0EE × 3378XE61E31 × 14X23716X5359331E9241 × 238566323E73EE11537XE996
7211 × 2EE × 35261 × 4159911 × X4728791 × 1EX866X721 × 273263674E × 1X5729993X7EX28E3E161 × 4X748X0X65EXX3943375X351994
734E × 111 × 251 × 1011 × 123EE × 15960E × 12478805981 × 50970299309727371 × 160605E10497012E4E × 8XX2X32378X50722348X7721XX5
745 × E × 11 × 1E × 25 × 75 × 175 × 1461 × 2181 × 3801 × 1X7X6335 × 754E2E41 × E0E0E0E0E1 × 113006390X1 × 64323X6685306022E8102E16XX12046E513132
7512E × 12X1 × 59E3E × 743494471 × 10E747484336X6E1 × 233553E3X2XX2X41 × 96E647633876838EE5E × 126E002945E107EX8205E888
767 × 11 × 17 × 27 × 31 × 51 × 111 × 131 × 2E1 × 471 × E61 × 1061 × 57E1 × E0E1 × 11111 × 15991 × 18787 × 3X891 × 1905201 × 7229231 × 7843701 × 91E0577131 × 10620610620611E1E3
7746E × 2X3E × 1E0411 × 69X3901 × 15163345X7587E × 118897E3050E0E2X6X0486585E × 68X1921E6396X5666EEE98X1E88911X711773
785 × 11 × 25 × 3E × 591 × 7231 × 78935EX441 × 523074X3XXE × 3266712021E531E1 × EE00EE00EE00EE00EE00EE00EE00EE00EE00EE00EE01XX1
79111 × 271 × 5201 × 509961 × 365E0031 × 3178744351 × 464069563E × 39478E3664E × 17362950433E533544947E563334X8075359X55301994
7X11 × X541 × 4176937 × 4235368181 × 832966217X8X111 × 89898362353285XE87E22X3EX7 × 16EE6202E02X5311278504010EX13001774
7E13E × 11111 × 1111111111111111111 × 8362843326116X29E1610998123X98060533261X67E8E8EX5X9820154X075944023X8E442
805 × 7 × 11 × 15 × 17 × 25 × 75 × 81 × 111 × 141 × 175 × 4401 × 4541 × EE01 × 1E601 × 75115 × 106X95 × 8E5281 × 1067281 × 146609481 × 1748E3674115 × 2792X26182722E6676819411X1X3
811111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
8211 × 157 × 46E × 2X3E × 7687 × 38E01 × 221X799347E672054XE87 × 5608XX9383413481532207 × 3257955345X23186304E321167X366X2593101992
83E × 1E × 31 × 111 × 291 × 1X01 × 3X891 × 368E51 × 2013881 × 754E2E41 × 16555E1X1 × 9703244140E09362XX4E5E4391 × 2E77660739X567436X957763E3X6111114
8453 × 11 × 25 × 85 × 841 × 1167 × E0E1 × 11111 × 2E0X1 × 36967 × 1277EE × 24727225 × 102X155X1 × 1768XX881 × 3185EE282985 × 9X06176590543EE × E64944E54364E0517155
8513X34E44X48463XE × 9XX69515E88505036X5X8481573922103434716847930357X727X45X490EX3015091808814E0270X096EE222
867 × 11 × 17 × 87 × 111 × 217 × 1587 × X9X9XE × 6E50611 × 126180EE0EE × 7605857409257 × 367XE43X31E761 × 24915461720E701 × 16EEX75X0381X76766164247X112124
872191 × 4361 × 30671E561 × 3780622808218059660EE × E536195X79679E64X9X7250756EEEE × 17440E0742X924600X5E1619X975XEXX14341666
885 × 112 × 25 × 45 × 67 × 75 × 175 × 485 × 18X31 × X8837 × 11E241 × 1E0411 × 69X3901 × 85653E5 × 6X78XX0721X7626395X1 × 127X6X3650767881EE41272X49EXE3X91927515143
8951 × 5E × 111 × 34E × 46E × 471 × 1561 × 2X3E × 57E1 × 11111 × 60E01 × 32XXE1 × 205812E × EX59849E × 1X2X6E6951 × E00E00EE0EE1 × 968035393909E245250919E5863772115154
8X11 × 8E × 51E × 1X11 × 17181 × 698X51 × 9X6X571 × 21XE0377 × 41X866326E31 × 191017735473010X59437791 × 18E73X9E7319678X467E056131195024481537EE4
8E60E3E × 17565X300E × 13E2717XX7627287E54X06115112E0E1707454800170172854268382989656X12601X667825E0E2299X1E53EE0261333
905 × 7 × 11 × 17 × 25 × 31 × 61 × 91 × 111 × 1X7 × 301 × 347 × E61 × 1061 × 2301 × EE01 × 1X761 × 3X891 × 129691 × 12387901 × 13X29831 × 1E807X62E61 × 27EX0644X61 × 683E6EE785X61 × 989457643023121215
911111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
92E × 11 × 1E × 237 × 3081 × 3767 × E0E1 × 11111 × 754E2E41 × E0E0E0E0E1 × 417569335X9871 × X78E8X09362X4X67 × 190E6445E1349XE4387 × X54106288E117883425268112124
93111 × 1398641 × 9E2X6732EE74552406X78E76247691 × E00E00E00E00E00E00E00E00E00E00E00E00EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE1441
945 × 11 × 15 × 25 × 75 × 81 × 95 × 157 × 175 × 46E × 481 × 1295 × 2X3E × 7687 × E9755 × 106X95 × 215415 × 2672288X41 × 15396X698X896281 × 415E02X229X8104401 × EEEE0000EEEE0000EEEE000119196
9516E × X03E47E088EE × 2E550E66X7197E × 24386950X07374EX744E × 1513E54025X9EX58802913E8329698X84X3999E5X739X6119863E9E1E016486291555
967 × 11 × 172 × 111 × 171 × 1XE7 × 4901 × 30351 × 55E81 × 630E1 × 127543624027 × 43404X297987 × 2376268253771 × 180218815260491 × 49535104455502961 × 111111111111111111115144
973E × 156E × 11111 × 78935EX441 × 523074X3XXE × 761556614942428313X650583XX7608656751269E9X82X657EX43057655E34E624346133866E6EE04805E662
985 × 11 × 252 × 4E × 8X5 × 123EE × 15960E × 163065 × 331887791 × 109E77147E205 × 28XE986X778834661 × 160605E10497012E4E × 15E4751582927483X81 × 348EX480E981E13E8E2113125
9931 × 111 × 3961 × 19491 × 3X891 × 1E0411 × 5XE48X1 × 69X3901 × 1064119E745041 × 729X1E8945E631 × 98717158913E7756714X091 × 658X50E23679083391136E114X12446110104
9X11 × 4E1 × 747 × 1092E × 4X82344661 × 146987541X202897 × 13167124X1E5E6E24E1 × 6600348E4E8666E8417207808371 × 991027128X344E58009X13377728X377816E995
9E17E × 46E × 9E1 × 2X3E × 67401 × X9X9XE × 7891X7E × 126180EE0EE × 1X6E953XX0X569X48034566217EXX38651087X821EEX4XX846X1X39151536560E1230267X513EX41995
X052 × 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 × 72066X5588687E7205E7998621124124232
X1E2 × 1E × 408E × 754E2E41 × 1E2347X16E × 273389373E × 26068098770838687570277297428517E3623E521 × 3097108E5XXEE91692504067248816X526X43908150E871985
X211 × 54991 × 6X3531 × 23E214X01 × 151540EX01 × 6E9704475971 × 25785880331329317729818949936575071 × 9X134140XE002419X65090184E86425024X210X79953X1884
X36E × 111 × 472488E21 × 4E2EX47X7863X18E5E18253377315E × E00E00E00E00E00E00E00E00E00E00E00E00E00E0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE0EE1551
X45 × 11 × 25 × 271 × 259441 × 50E237 × 365E0031 × 464069563E × 39478E3664E × 3012X78X7X0X1 × 3373558414615602861 × 5977650E39345191054X8941 × 2221710X303X15671780413X711114
X518E × 7E5E × 11111 × 5887E × 1277EE × 9X06176590543EE × 19906116507647451X176848466145277E42E8274EX793924127713152370055529E473202388800E17E01370E774
X672 × 11 × 17 × 31 × 37 × X7 × 111 × 157 × 46E × E61 × 1061 × 2X3E × 7687 × 9X17 × 3X891 × 76E077 × 22671E17 × 5X9815491 × E00E00EE0EE1 × 1062060EEE001062061 × EEE000000EEE000000EEEEEE000EEEEEE0011X194
X71524X46E41E × XE40396414558012072859E47584E1E × X01E6709327152983887984680242642205X31132061454580217E479100E22X553675016298052383851333
X85 × 11 × 15 × 25 × 75 × 81 × 175 × 541 × 18X81 × 3E055 × 75115 × 106X95 × 1E395315 × 1748E3674115 × 491E191565059401 × 22E6E3E4614X2X0E739X6493730X81 × 2344208779XX6258285840X8052004X8115155
X9111 × 2EE × 1326391 × 4159911 × 273263674E × 4X748X0X65EXX3943375X351 × 8824771150225312417191041661931977572125372E14654697446827187452864604E8535221772
XX112 × 67 × XE × E0E1 × 11111 × 18X31 × 19X87 × X8837 × 1E0411 × 69X3901 × 158X0217 × 45E152651 × 328X222960EE4296E × X003996X2X7736694E501 × 49X2886048599849242952388618X3901946114133
XE19E × 72018384965970451E067536910118X32E1123514585X8208319E4E223E82731346976E3357E3X060X5525748EXE5590563737X2842E7804E892129EE603X70E222
E05 × 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 × 4622E8098X921X0916X79XXX123234
E146E × 2X3E × 32961 × X2645EEE × 1111111111111111111 × 1E199E5X1154899267903XX31 × 20X584263846XE267E0734E2X71X15426X58EEX131091169959376E267X50562XE0E379E774
E211 × 219X7 × 926337 × 13286641 × 3176XE592E × 118357X16417E44E × 2X7304X3E254X17927452584836E1769951 × 68589X43X7633909564843X29E049E4413X711E90945851378E91E01883
E331 × 51 × 111 × 471 × 57E1 × 11111 × 15991 × 3X891 × 129691 × 1905201 × 7229231 × 7843701 × 9894576430231 × 3082666E0496E4611X161 × EE16650679438E4210231 × 3E484218E44X9828444735035540E3114143
E45 × 11 × 25 × 75 × E5 × 175 × 1587 × X9X9XE × 336E8XX001 × 126180EE0EE × 1349X9E47X1 × 7605857409257 × 93503726E44887X0575721 × 39X756215584887E1214176X2007106E6730624521EX07194893512123
E5278888E × 6559E89566E16861 × 35X0822EX99E414205806454EE0917851692858E × 27805997X962266E9XX8X3X504E199E7491540342X14E313791X72X00627776026XX68E1X11444
E67 × 11 × 17 × 3E × E7 × 111 × 1E1 × 591 × 4X57 × 7231 × 7671581 × 374594E7 × 78935EX441 × 523074X3XXE × 3266712021E531E1 × 29E724E34313E08941 × 76E45076E4545076E45077 × 3X7E21516142789517140EE6X7116166
E76442E324EE × 5759X3929EE8E112931 × 39X37X50E66476X58688XX15390622XX480X471396596E × 11950E73490699E9X211171624E426E56905E3846326383X686767093X2281E51444
E852 × 11 × 25 × 5E × 157 × 34E × 46E × 481 × 2X3E × 7687 × E0E1 × 11111 × 32XXE1 × 205812E × 200X2121 × 24727225 × EX59849E × 201347X741 × 2672288X41 × 130982376E76481 × 247X950907970707X1 × 10EEEXXXE011110EXXXE000111E1X4
E9111 × 120EX31 × 400177X625E2E91 × 832966217X8X111 × 16EE6202E02X5311278504010EX13001 × 24168717605137390789685298X0709447047E8957264038813319910978876E242X5XE1663
EX11 × 2E61 × E411 × 54820E8449361 × 207E168E81X24231305XEX70916E7325811X6394X492811XE4311E2351 × 48214E17105E9X31322076E52260904296018248289388E3E03E1E49225X961662
EEE × 1E × 1E0411 × 69X3901 × 754E2E41 × 259EE7EX1 × 42EXEEE90E70755X2317X981 × 105E2X559298E37XX0X7E6E19729857X29748931432081246479069E8512X716636446E7E2X1062948E569051883
1005 × 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 × 323X555614903716989X1980126264

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>1d(10)), where Φd(10) is the dth cyclotomic polynomial evaluated at 10.

Properties Edit

  • 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<jn 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 nRj(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(mn) = gcd(m - nn) 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(mn) = 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 Edit

Positive bases Edit

  • 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 Edit

  • 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 primes base b Edit

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 known length of repunit primes in base b
−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, ...
−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, ...
−67 3, 8E, 321, 34E, 11E7, ...
−66 3, 7, 27, 471, 2535, ...
−65 31, 225, ...
−64 3, 5, 13E, 1X5, 11771, ...
−63 5, 6E, 3717, ...
−62 2, 11, 27, 31, 91, X087, ...
−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, ...
−52 2, E, 25, 11E, 221, 9707, 1X48E, ...
−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, ...
−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, ...
−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, ...
−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, ...
−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, ...
E 15, 17, 61, E7, 637, 112E, 1211, 2941, 2E95, 6357, E801, 12205E, ...
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, ...
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, ...
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, ...
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, ...
38 5, 27, 11E, 4X1EE, ...
39 17, 45, 11E, 1E07, 6621, 17X67, ...
3X 2, 7, 17, 57, 157, 301, 14E1, 16X7, E377, ...
3E X7, X511, 1XE1E, ...
40 17, 1X5, 251, 27E, 907, 8847, ...
41 (none)
42 3, 5, X7, E7, 24E, 471, 1337, 3935, ...
43 2545, 18477, ...
44 2, 87, 195, 2545, 399E, ...
45 E, 27, 35, XXE, 12XE7, ...
46 3, 285, 9655, X76E, 3E887, ...
47 15, 35, 3E, 107, 59E, 138E, 1E0E, 2127, 3335, 1869E, 48E57, ...
48 7, 111, 1257, 1471, 29537, ...
49 3, 15, 91, 107, 157, 471, 9997, 10905, ...
4X 2, 35, 1425, 33325, ...
4E 3, 11, 33E, 710E, ...
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, ...
54 (none)
55 17, 25, 447, ...
56 2, 3, 7, 17, E685, ...
57 17, 267, X3E, 1E2E, 26XE, 601E, 7917, ...
58 5, 7, 8E, 105, 1727, ...
59 3, 51, 1457, 2085, 4971, 516X5, ...
5X 2, 25, 4E, 391, 535, 705, 6885, 13EX7, ...
5E 3, 27, 35, 111, XEE, 15E9E, 27X5E, 35837, 73261, ...
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, ...
66 2, 3, 85, 195, 1165, 32X31, ...
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, ...
74 2, 51, 401, 21X7, 1124E, 1E6EE, ...
75 3, 7, 37, 3E, 5E, 91, 3E7, 6E17, 24E85, ...
76 3, 17, 81, 3021, ...
77 2685, E7E1, ...
78 307, 7635, 11151, 218E7, ...
79 7, 2X07, ...
7X 5, 11, 31, 1051, 20X5, ...
7E 7, 377, 5457, 609E, 678E, ...
80 2, 1E27, 23127, ...
81 15, 31, E91, ...
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 ppositive bases such that Rp is primenegative bases such that Rp is prime
32, 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, ...
52, 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, ...
72, 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, ...
E5, 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, ...
112, 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, ...
152, 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, ...
172, 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, ...
1EX, 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, ...
256, 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, ...
272, 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, ...
3151, 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, ...
3512, 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, ...
3713, 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, ...
3E5, 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, ...
4520, 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, ...
4E17, 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, ...
512, 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, ...
573X, 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, ...
5E3, 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, ...
61E, 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, ...
671X, 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, ...
6E35, 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, ...
752, 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, ...
8110, 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, ...
851X, 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, ...
873, 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, ...
8E2, 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, ...
9110, 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, ...
9572, 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, ...
X72, 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, ...
XE7, 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, ...
E511, 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, ...
E7E, 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, ...
1055, 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, ...
10725, 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, ...
11148, 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, ...
11726, 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, ...
11E38, 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, ...
12550, 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, ...
12E214, 33X, 40X, 666, 674, 693, 7XX, ...E, 47, 193, 201, 229, 32E, 605, 866, 9E7, X07, E03, ...
1315, 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, ...
13E62, 15X, 2X8, 339, 417, 474, 4X3, 4E4, 51E, 650, 7E8, 872, ...2, 30, 64, 37E, 3EE, 422, 7X8, 846, X40, X47, E50, ...
1419X, 211, 346, 3X2, 451, 481, 514, 756, 83X, XX8, ...10, 2X0, 353, 426, 469, 644, 705, 791, 823, 871, X74, X9X, ...
14529, 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, ...
147110, 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, ...
1573X, 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, ...
167133, 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, ...
16E60, 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, ...
171426, 505, 52X, 5E6, 672, XEX, E19, ...5, E, 47, 85, 14X, 310, 334, 5X3, 704, 708, 749, 7X1, X93, E04, E36, ...
175422, 982, 985, ...349, 556, 770, 813, X70, E79, ...
17E167, 198, 267, 336, 3E0, 5EX, 9E3, ...59, X6, 159, 34X, 499, 5X5, X57, XE7, E5E, ...
18197, 117, 167, 1X1, 1X6, 236, 495, 5X9, 706, 71X, 9E3, XX4, ...E5, 162, 24X, 43X, 487, 49X, 540, 61X, 670, 70X, 760, E48, EE6, ...
18E31, 186, 1X3, 436, 659, 796, ...14, 37, 163, 218, 30E, 3E3, 460, 523, 577, ...
19544, 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, ...
19E88, 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, ...
1X535, 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, ...
1X76, 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, ...
1E1242, 335, 451, 664, 665, 696, E19, ...22, 37, 60, 148, 15E, 276, 2XX, 413, 5X0, 621, 677, 688, E97, ...
1E5161, 312, 426, 436, 55X, 600, X98, EE2, ...3, 101, 399, 562, 709, 8E0, ...
1E711, 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, ...
205E4, 284, 333, 8XE, ...X, 6X, 84, 97, X2, 345, 62E, 7X7, X15, ...
217164, 177, 224, 256, 27E, 325, E98, ...14, 7X, 255, 342, 36E, 645, 825, X96, ...
21E116, 404, 695, 845, XX5, XX9, ...155, 246, 30E, 351, 547, 598, 651, 726, 901, X86, X93, E18, ...
2212E, 87, 1X7, 327, 435, 924, ...2, 52, 147, 172, 20X, 376, 634, 694, ...
225X, 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, ...
237162, 187, 353, 387, 67X, 790, 906, 940, 94E, X0E, X71, X82, ...1E, 3X, 115, 212, 245, 367, 370, 587, 594, 649, E23, E65, ...
24117, 85, 93, 24E, 299, 348, 3E8, 420, 889, 949, X10, XE4, ...1X9, 211, 268, 741, 968, ...
24E22, 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, ...
25133, 40, 404, 434, 462, 519, X7E, E47, ...324, 454, 473, 646, 657, 69X, 794, 801, 887, E54, ...
25510, E2, 11X, 149, 453, 512, 580, 997, ...1X, E5, 17X, 588, 684, 6X7, 85E, 892, 92X, 97E, ...
25E1X, 150, 2E6, 392, 3X0, 789, ...3, 51, 70, 262, 34E, 412, 480, 746, 772, 949, ...
26757, 52E, 621, 738, 7E9, 876, 915, 929, X00, X87, E50, EX4, ...30, 107, 1XX, 439, 600, 628, 677, 678, 9E2, X02, ...
271X0, 11E, 84E, 880, 929, X20, X40, ...24, 278, 3X1, 551, 737, 741, 832, X44, ...
277143, 3XX, 4X1, 753, 7E7, 868, ...235, 236, 288, 2E0, 3E9, 400, 459, 5XX, 6E2, 90X, 923, EX2, ...
27E40, 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, ...
28546, 89, 2X2, 2EX, 317, 4E7, 546, 559, 781, 983, 9E5, X41, EE1, ...59, 392, 445, 521, 5X0, 99X, 9E4, ...
291327, 443, 4X9, 781, 817, ...72, 77, 198, 205, 312, 469, 916, ...
29532, 168, 24E, 342, 34X, 728, 736, 952, 9X2, XX3, E32, ...1X7, 345, 647, 709, 727, 822, 894, 952, 9E0, E28, ...
2X135, 351, 46E, 735, ...290, 2E9, 35E, 754, 831, 911, XE7, E29, ...
2XE15, 258, 66X, ...24, 129, 1XX, 3E8, 540, 581, ...
2E1574, 597, 5X2, 9EX, ...6E, 1X8, 235, 623, 786, 8X6, 914, 9X7, E50, EE1, ...
2EE298, 534, 6E8, 926, 949, ...212, 248, 47X, 555, 599, 817, 87E, EX2, ...
3013X, 2E5, 3X3, 5X5, 609, 657, 786, E09, E48, E95, ...155, 202, 406, 453, 455, 522, 553, 711, 780, X80, ...
30764, 78, 165, 502, 529, 922, X88, E28, ...E, 68, 14E, 1EX, 285, 666, 75X, 940, 954, X87, XX5, E14, ...
30E561, 768, ...210, 651, 6E6, ...
31532, 90, 10E, 155, 167, 483, 531, ...113, 131, 270, 31X, 562, 65X, 801, 840, 881, ...
32124, 29E, 2X6, 501, X6E, E73, ...67, 250, 307, 473, 484, 49E, 822, E59, ...
32515E, 170, 18E, 2X6, 459, 463, 501, 525, 6X1, 914, X11, ...27, 84, 3E9, 400, 4E0, 4E8, 613, 718, 751, 7EX, 87E, 97E, 9E5, E6X, ...
32731, 240, 464, 594, 801, 811, 836, 94X, EX3, ...237, 36X, 750, 906, X71, ...
32E178, 483, 553, 5XX, 614, 682, 75E, E7E, ...44, 73, XX, 456, 521, 91X, ...
33E4E, 214, 296, 358, 515, 631, 7E9, 850, E72, EX9, ...128, 13E, 499, X47, X97, E6E, ...
34713, 195, 1EX, 2X8, 438, 4E7, 592, 63E, 67X, ...3, 350, 73X, E2E, E52, ...
34E36X, 917, ...24, 67, 7E8, 9XE, ...
357198, 4X7, 992, ...161, 215, 222, 536, 695, ...
35E356, 389, 397, 424, 4E9, 588, 62E, 687, ...12, 2E, 37, 59, 612, 873, 8E0, X38, X65, ...
3656, 72, 132, 600, 836, X95, E08, ...2X2, 846, 905, ...
3752, 31, 325, 402, 436, 5X2, 6E2, 739, 7X3, ...190, 3E3, 47X, 48X, 6X6, 6E6, 922, 968, X05, E19, ...
3777E, 572, 676, ...4EX, 650, 6XX, 9EE, X1X, ...
3913, 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, ...
397335, 37E, 480, 785, ...9, 26, 6E, 185, 3X9, 594, 784, 9EE, X78, E39, E40, ...
3X52X9, 6X0, 824, 921, 944, ...E, 19X, ...
3XEX3, 192, 42E, 544, 632, 801, 8E1, 984, ...300, 347, 750, 776, 884, X1E, X83, E16, E64, EEX, ...
3E526, 96, 472, 536, 568, 832, 854, XX8, ...125, 192, 276, 661, X34, ...
3E775, 103, 243, 3X9, 655, 771, 821, ...79, 242, 864, 955, 971, 9X7, E83, ...
40174, E8, 113, 147, 979, X44, X47, E95, ...3, E6, 263, ...
40E178, 476, 59X, 5X3, 625, 746, 781, EX5, EX6, EXX, ...1E, 140, 149, 381, 482, 672, 72E, ...
41564, 23E, 566, X77, X91, ...149, 32X, 662, ...
41EX4, E1, 279, 301, 382, 659, 8E6, X91, ...128, 357, 404, 433, 813, ...
421...79, 10X, 17E, 375, X14, E79, ...
4272, 571, 691, 757, 982, E57, E90, ...439, 892, 89E, ...
431140, 155, 249, 28E, 455, 458, 61X, 922, ...7X, 184, 197, 210, 22E, 473, 672, ...
435137, 215, 44E, 482, 982, X76, ...2E, 1X0, 317, 626, 804, X19, ...
4375, 16X, 22E, 314, 5E1, ...10X, 1X7, 555, 762, X46, X77, ...
44733, 55, 475, X03, E7X, ...17, 339, 624, 830, X07, E75, ...
455897, ...X, ...
45713X, 452, 672, 709, 821, X25, ...56, 340, 355, 4E0, 529, 90E, 984, ...
45E229, 65E, 809, E23, ...124, 35E, 5X4, 614, 838, 839, ...
46520, ...56E, 86E, ...
46E67, 244, 653, ...X59, E64, ...
47120, 42, 49, 510, 565, 7E8, 936, 972, 9X8, X68, X7X, ...66, 117, 732, ...
48186, 13X, 2E5, 5E7, 905, ...88, 351, 352, 620, 993, E26, ...
48585, 544, 722, 91X, 969, X90, ...256, 2E9, 405, 659, 770, 969, X30, X34, ...
48E358, 359, ...201, 33X, 381, 677, 67X, 71E, 908, X9E, ...
49792, 677, 784, E64, E72, ...35, 502, 786, ...
4X510, 94, 162, 193, 477, 57X, 6X1, 755, X44, E89, ...2, 69, 233, 495, 535, 537, 599, 996, 9E7, ...
4E196, 1X7, 399, 585, 897, E26, ...18, 2E, 36, 74, 163, 781, 8E2, EE7, ...
4EE1E7, 27X, ...2EX, 6E3, X26, ...
5076E8, X70, ...140, 1X7, 31X, 458, X06, ...
5113E2, 888, 976, E86, ...16, 402, ...
51763, 3X3, 653, E55, E99, ...X3, 289, 2X4, 3XX, ...
51E292, 387, ...684, ...
52734, 15E, 397, 6E8, 737, 760, ...248, 2E5, 474, 813, ...
53152, 511, 893, X17, ...58, 393, ...
5355X, 522, 894, ...X44, EE4, ...
54151, 322, 3X4, 477, 783, X43, ...105, 112, 321, 456, 638, 809, 98E, ...
5458X4, ...9, 58, 374, 411, 454, 888, X39, ...
557268, 286, 422, X41, ...307, 354, 70E, 774, 90E, X38, XX2, XX9, ...
565339, 949, X85, ...18, 9E, 345, 446, 597, 8XX, 986, ...
575582, E6X, ...10X, 111, 128, ...
577246, 311, 410, 562, 933, X13, XE4, ...6, 91, 272, 432, 539, XE3, ...
585161, 4X4, 580, 64E, 6E1, 7XX, ...28E, 426, 461, 580, 831, 85X, 90E, X43, ...
587120, 909, 9EX, ...27E, 459, 574, ...
58E9E, 4EE, 504, 8E4, 9E3, ...12E, 178, 447, 57E, 965, ...
59114X, 1X7, 529, 73X, ...26, 69, 305, 337, 581, ...
59E47, 210, 3X6, 3E3, 430, 530, X30, ...88, 273, 44X, 64X, 880, ...
5E12EX, ...181, 318, ...
5E51X, 102, 858, 955, ...22, 2E, 167, 18X, 1X6, 965, E52, ...
5E7306, 6X3, 792, ...372, 589, 679, 893, 8E0, X5X, X86, E72, ...
5EEX83, ...204, 22E, 490, 896, 973, 9EX, ...
611601, 682, ...369, ...
6151XE, 49E, 592, 5X2, 897, E94, ...267, 412, 985, XX7, ...
61711, 192, 398, 990, X99, ...155, 669, X37, ...
61E244, 3E2, 427, 684, 870, XX5, ...310, 46X, 473, 482, 966, X23, XX5, ...
637E, 338, E04, ......
63E12X, 520, 575, 669, ...367, XEX, ...
647638, 880, ...11, 657, 662, 744, 974, ...
6555, 58E, 653, ...374, 927, ...
661590, 609, 634, ...50, 146, 402, 703, ...
665180, 2X9, 52X, ...16, 66E, E61, ...
66E190, 471, 889, 8E4, E39, ...28X, 430, ...
675174, ......
687195, 2E7, 527, 993, ...15, 196, 527, 656, 710, 896, 928, ...
68E162, 594, X20, ...70, X74, X79, ...
695E35, E37, ...921, ...
69E103, 2E2, ...8E, 227, 355, 362, 462, 9X7, E56, ...
6X711, E1, 2X7, 625, 904, ...46, 4E, 957, ...
6E1359, 3EE, 413, 466, 623, 624, X48, ...167, 737, 9X8, ...
701E5, 236, 492, 839, 849, 9X6, ...9, 74X, XX5, ...
7055X, E79, ......
70E1E4, 204, 897, E12, ......
71111, 531, 75E, 991, 9E1, X07, E90, ...737, E14, ...
71EX, ...24, 69, 88, 856, ...
721798, 870, ...69, 972, 978, E00, ...
727231, 320, 404, 610, 836, ...6X6, ...
7356, 836, 83E, 924, ...144, 191, 212, 500, 580, E56, ...
737867, E17, ...47X, 494, E12, E7X, ...
74517, 16E, 2X3, 685, 749, 8E8, E85, ...27, X7, 18X, 379, 52E, ...
747X58, X59, E52, ...32, 48, 109, 210, 288, ...
75172, 55X, 731, 99X, ...152, 81E, ...
767116, 5EX, 729, ...70, 143, 364, 536, 60X, ...
76E3, 276, 4X7, ...13, 444, 44E, ...
77117X, 27E, 380, ...57E, 664, E67, ...
775474, 487, ...16, 163, 210, 448, 663, X84, ...
77E240, 314, ...75, ...
7851X3, 257, 527, 701, 716, 852, ...118, 42E, 516, 95X, ...
79168X, 881, 973, ...171, 630, E6E, ...
797X71, ...2E5, 536, ...
7X1...47, 2X2, 805, 910, 955, X58, E6E, E84, ...
7EE240, ...11, 4XE, 692, 730, 9XE, ...
801286, 6E3, 976, ...42, 13X, 637, 967, 9XE, ...
80E288, ...18, 13E, 292, 397, 579, 673, 784, 865, 8E6, X80, ...
817E3, 29X, 36X, X06, X42, ...E2, 669, ...
82585, E1, 732, 891, X39, ...7X5, ...
82E411, 68E, ...89, 2X9, 547, X05, ...
83511, 91, 221, 226, 55X, 6X0, X27, ...254, 36X, 8X8, ...
841104, 1E6, 217, 34E, 61E, ...257, ...
85111E, 1X7, 558, 876, E70, ...X5E, ...
85531X, 429, 459, ...34, 10X, 173, 256, 40X, 45E, 8X5, ...
85E250, 274, 5X6, ...46X, 5EE, ...
865154, 440, ...12, 731, 809, ...
867EX, 639, 6E6, ...253, 49X, 538, 61E, 8E8, ...
871122, 2X8, 805, X90, ...235, 240, 98X, E7E, ...
881189, 442, 569, ...21, ...
88E865, ...326, 63X, 653, ...
8X534, 162, ...169, 45X, ...
8X72, 727, ...6X, 26X, 4EX, X32, ...
8XE470, 540, 671, X7E, E03, EX2, ...417, 484, ...
8E511X, 786, 91E, ......
8E72X4, ...X16, ...
9012E, 561, ...680, 9X4, ...
905806, ...110, 11E, 650, 94X, 95E, ...
90740, 672, 992, ...E0, 208, 342, 373, 556, 960, X80, ...
90E163, 4X1, ...128, 570, 5E7, 877, X74, ...
91E36, 2X8, 38X, X50, ...206, 654, ...
9212E4, 6X5, 801, ...47, 450, 677, 824, ...
9275EE, ...456, 615, 7E0, ...
9554E9, 803, 897, E9X, ...7XX, ...
95E3, 452, 487, 650, ...257, 799, XX0, ...
965384, 453, 512, ...74, 18X, 205, 4X4, ...
971176, 17X, 4E7, 8E3, ......
98713E, 65E, E75, E85, ...1E2, 43X, 893, 982, XE1, XEX, E11, ...
99597, 83E, ......
9X724, E3X, ...240, 251, E43, E76, ...
9XE...510, ...
9E1572, 5X5, ...415, 71X, 738, ...
9E5113, 196, ...362, 385, 78X, ...
9EE293, 865, ...102, 4X2, 680, X3E, X7E, ...
X075X9, ...4X, 2E4, 488, 5E5, 878, 9X6, ...
X0E...924, ...
X11297, ...560, 74E, 878, ...
X17987, ...16X, 259, 491, 605, ...
X273X6, 916, ...E9, 321, 623, X91, ...
X35...41, 98, E78, ...
X37312, 747, ...10, 115, 30E, 671, 71X, ...
X3E18, 57, 114, EXX, ...132, 543, 937, ...
X41423, X43, ...296, 95X, E81, ...
X452X, ......
X4E25X, 356, 53E, 855, ...2E, X1, ...
X5E91E, ...XE9, ...
X6E316, 4EE, ...946, ...
X774E7, 579, 691, ...1EX, 687, ...
X876X4, 8XE, X8X, ...26E, 719, E30, ...
X9187, 228, 404, XX3, ......
X95...538, 596, 9X3, ...
X9E996, ......
XX7843, ...643, ...
XXE45, 228, 32E, ...241, 32E, 398, ...
XE7725, 957, XX3, E13, ...3, 89X, ...
XEE5E, 154, 42X, 57E, E52, ......
E11465, 668, 7X6, ...32, 483, ...
E15852, ......
E1E...351, 5E8, ...
E21132, 340, 396, 3X0, 558, 732, ...238, 413, 486, 914, ...
E25264, 433, 63X, 80X, E47, ...3E1, ...
E2E328, ...7, ...
E311E1, 275, 515, ...5X8, 692, 6XX, E50, ...
E373, 603, X45, ...89, 3X3, 91E, XE6, ...
E45117, 54E, ...36, 363, 425, 561, ...
E61...166, E1X, ...
E67222, 453, 950, EE6, ...3, 2X0, 478, 674, 782, ...
E6E33X, 90E, ...50, 687, 697, ...
E713X4, X92, ...16X, X10, ...
E9181, X0, X3, ...150, 962, X37, ...
E9594, 45X, 988, ...332, 576, ...
E977, 127, 268, 2XE, 404, 527, 899, X4X, E73, ...289, 618, 779, 955, E52, ...
EX52E8, 40E, 424, 575, 621, ...2, 21, 542, X85, ...
EE592, E8, 222, 521, 834, ...58X, ...
EE7375, 629, ...8XE, 970, ...

The generalized repunit conjecture Edit

A conjecture related to the generalized repunit primes:[1][2] (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many repunit primes for all bases

$ b $ ) For any integer

$ b $ , which satisfies the conditions:

  1. $ |b|>1 $.
  2. $ b $is not a perfect power. (since when$ b $ is a perfect$ r $th power, it can be shown that there is at most one$ n $ value such that$ \frac{b^n-1}{b-1} $ is prime, and this$ n $ value is$ r $ itself or a root of$ r $)
  3. $ b $is not in the form$ -4k^4 $. (if so, then the number has aurifeuillean factorization)

has generalized repunit primes of the form

$ R_p(b)=\frac{b^p-1}{b-1} $

for prime

$ p $ , the prime numbers will be distributed near the best fit line

$ Y=G \cdot \log_{|b|}\left( \log_{|b|}\left( R_{(b)}(n) \right) \right)+C, $

where limit

$ n\rightarrow\infty $ ,

$ G=\frac{1}{e^\gamma}=0.68\mathcal{X}25104\mathcal{E}904... $

and there are about

$ \left( \log_e(N)+m \cdot \log_e(2) \cdot \log_e \big( \log_e(N) \big) +\frac{1}{\sqrt N}-\delta \right) \cdot \frac{e^\gamma}{\log_e(|b|)} $

base

$ b $ repunit primes less than

$ N $ .

  • $ e $is the base of natural logarithm.
  • $ \gamma $is Euler–Mascheroni constant.
  • $ \log_{|b|} $is the logarithm in base$ |b| $
  • $ R_{(b)}(n) $is the$ n $th generalized repunit prime in base$ b $ (with prime$ p $)
  • $ C $is a data fit constant which varies with$ b $.
  • $ \delta=1 $if$ b>0 $,$ \delta=1.6 $ if$ b<0 $.
  • $ m $is the largest natural number such that$ -b $ is a$ 2^{m-1} $th power.

We also have the following 3 properties:

  1. The number of prime numbers of the form$ \frac{b^n-1}{b-1} $ (with prime$ p $) less than or equal to$ n $ is about$ e^\gamma \cdot \log_{|b|}\big(\log_{|b|}(n)\big) $.
  2. The expected number of prime numbers of the form$ \frac{b^n-1}{b-1} $ with prime$ p $ between$ n $ and$ |b| \cdot n $ is about$ e^\gamma $.
  3. The probability that number of the form$ \frac{b^n-1}{b-1} $ is prime (for prime$ p $) is about$ \frac{e^\gamma}{p \cdot \log_e(|b|)} $.


Demlo numbers Edit

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, ...


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