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

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):

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

where ${\displaystyle \Phi_d(x)}$ is the ${\displaystyle d^\mathrm{th}}$ cyclotomic polynomial and d ranges over the divisors of n. For p prime,

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

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 ${\displaystyle \Phi_2(x)}$ and ${\displaystyle \Phi_3(x)}$ and ${\displaystyle \Phi_4(x)}$ and ${\displaystyle \Phi_6(x)}$ and ${\displaystyle \Phi_{10}(x)}$ are ${\displaystyle x + 1}$ and ${\displaystyle x^2+x+1}$ and ${\displaystyle x^2+1}$ and ${\displaystyle x^2-x+1}$ and ${\displaystyle x^4-x^2+1}$, 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 (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 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)

Helmut Zeisel found that p = 2^(k-1) + k is prime if k = the repunit 1111 (1111 is exactly the smallest composite repunit number), the repunit 1111 can be factored as 1*5*11*25, and 5=2*1+3, 11=2*5+3, 25=2*11+3, such numbers are called Zelsel number, the famous number 1001 is also Zeisel number.

## 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.
• 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 × 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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>1d(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<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

### 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, 99887, E6411, ...
−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, 997X5, ...
−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, 11672E, ...
−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, 509XXE1, ...
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, 3E5791, ...
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, 234E71, ...
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, 1430X1, ...
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, 109221, ...
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 ${\displaystyle b}$) For any integer ${\displaystyle b}$, which satisfies the conditions:

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

has generalized repunit primes of the form

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

for prime ${\displaystyle p}$, the prime numbers will be distributed near the best fit line

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

where limit ${\displaystyle n \rightarrow \infty}$, ${\displaystyle G=\frac{1}{e^\gamma}=0.68\mathcal{X}25104\mathcal{E}904...}$ and there are about

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

base ${\displaystyle b}$ repunit primes less than ${\displaystyle N}$.

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

We also have the following 3 properties:

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

## 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, ...

1. Deriving the Wagstaff Mersenne Conjecture
2. Generalized Repunit Conjecture
3. Template:Harvnb, Template:Harvnb
4. Template:MathWorld