In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit.
A repunit prime is a repunit that is also a prime number.
In the sections below, R_{n} is the repunit with length n, e.g. R_{10} = 111111111111.
Repunit prime Edit
R_{n} is known to be prime for n = 2, 3, 5, 17, 81, 91, 225, 255, 4X5, and R_{n} is probable prime for n = 5777, 879E, 198E1, 23175, 311407. (Note that 879E is the only known such n ends with E)
R_{XE}/19E is a X8-digit prime (XE is the only known prime p such that R_{p}/(2p+1) is prime).
R_{141} has two 60-digit prime factors, and they are very close.
If _{n} is composite, then R_{n} is also composite (e.g. 2E = 5 × 7, and R_{2E} = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001), however, when n is prime, R_{n} may not be prime, the first counterexample is n=7, although 7 is prime, R_{7} = 1111111 is not prime, it equals 46E × 2X3E.
Theorem Edit
If p is prime other than E, then every prime factor of R_{p} is = 1 mod p. (e.g. both 46E and 2X3E are = 1 mod 7)
If p is Sophie Germain prime other than 2, 3 and 5, then R_{p} is not prime, since R_{p} must be divisible by 2p+1. (e.g. 1E|R_{E}, 3E|R_{1E}, 4E|R_{25}, 6E|R_{35}, 8E|R_{45}, 11E|R_{6E}, 12E|R_{75}, 16E|R_{95}, 19E|R_{XE})
If p is prime other than 2, 3 and E, then p divides R_{p}_{-1}, however, some composite numbers c also divide R_{c−1}, the first such example is 55, which divides R_{54}
For prime p other than 2, 3 and E, the smallest integer n ≥ 1 such that p divides R_{n} is the period length of 1/p, e.g. none of 1, 11, 111, 1111 and 11111 is divisible by 17, but 111111 is, and the period length of 1/17 is 6: 1/17 = 0.076E45
All repunit composite with prime length except R_{E} are Fermat pseudoprime (also Euler pseudoprime, Euler-Jacobi pseudoprime and strong pseudoprime) base 10.
If n is Fermat pseudoprime base 10, then R_{n} is also Fermat pseudoprime base 10 (thus, there are infinitely many Fermat pseudoprimes base 10).
Factorization of dozenal repunits Edit
(Prime factors colored red means "new factors", i.e. the prime factor divides R_{n} but not divides R_{k} for all k < n)
R_{1} = 1
R_{2} = 11
R_{3} = 111
R_{4} = 5 × 11 × 25
R_{5} = 11111
R_{6} = 7 × 11 × 17 × 111
R_{7} = 46E × 2X3E
R_{8} = 5 × 11 × 25 × 75 × 175
R_{9} = 31 × 111 × 3X891
R_{X} = 11 × E0E1 × 11111
R_{E} = E × 1E × 754E2E41
R_{10} = 5 × 7 × 11 × 17 × 25 × 111 × EE01
R_{11} = 1E0411 × 69X3901
R_{12} = 11 × 157 × 46E × 2X3E × 7687
R_{13} = 51 × 111 × 471 × 57E1 × 11111
R_{14} = 5 × 11 × 15 × 25 × 75 × 81 × 175 × 106X95
R_{15} = X9X9XE × 126180EE0EE
R_{16} = 7 × 11 × 17 × 31 × 111 × E61 × 1061 × 3X891
R_{17} = 1111111111111111111
R_{18} = 5^{2} × 11 × 25 × E0E1 × 11111 × 24727225
R_{19} = 111 × 46E × 2X3E × E00E00EE0EE1
R_{1X} = E × 11 × 1E × 754E2E41 × E0E0E0E0E1
R_{1E} = 3E × 78935EX441 × 523074X3XXE
R_{20} = 5 × 7 × 11 × 17 × 25 × 75 × 111 × 141 × 175 × EE01 × 8E5281
R_{21} = 11111 × 1277EE × 9X06176590543EE
R_{22} = 11^{2} × 67 × 18X31 × X8837 × 1E0411 × 69X3901
R_{23} = 31 × 111 × 3X891 × 129691 × 9894576430231
R_{24} = 5 × 11 × 25 × 157 × 46E × 481 × 2X3E × 7687 × 2672288X41
R_{25} = 4E × 123EE × 15960E × 160605E10497012E4E
R_{26} = 7 × 11 × 17 × 27 × 51 × 111 × 2E1 × 471 × 57E1 × E0E1 × 11111 × 18787
R_{27} = 271 × 365E0031 × 464069563E × 39478E3664E
R_{28} = 5 × 11 × 15 × 25 × 75 × 81 × 175 × 75115 × 106X95 × 1748E3674115
R_{29} = E × 1E × 111 × 368E51 × 2013881 × 754E2E41 × 16555E1X1
R_{2X} = 11 × 1587 × X9X9XE × 126180EE0EE × 7605857409257
R_{2E} = 5E × 34E × 46E × 2X3E × 11111 × 32XXE1 × 205812E × EX59849E
R_{30} = 5 × 7 × 11 × 17 × 25 × 31 × 61 × 111 × E61 × 1061 × EE01 × 3X891 × 1E807X62E61
R_{31} = 1398641 × 9E2X6732EE74552406X78E76247691
R_{32} = 11 × 1XE7 × 4901 × 127543624027 × 1111111111111111111
R_{33} = 111 × 19491 × 1E0411 × 5XE48X1 × 69X3901 × 1064119E745041
R_{34} = 5^{2} × 11 × 25 × 35 × 75 × 175 × 375 × E0E1 × 11111 × 62041 × 1X7X9741 × 24727225
R_{35} = 6E × 472488E21 × 4E2EX47X7863X18E5E18253377315E
R_{36} = 7^{2} × 11 × 17 × 37 × 111 × 157 × 46E × 2X3E × 7687 × 9X17 × 76E077 × E00E00EE0EE1
R_{37} = 2EE × 4159911 × 273263674E × 4X748X0X65EXX3943375X351
R_{38} = 5 × E × 11 × 1E × 25 × 1461 × 2181 × 3801 × 754E2E41 × E0E0E0E0E1 × 113006390X1
R_{39} = 31 × 51 × 111 × 471 × 57E1 × 11111 × 15991 × 3X891 × 1905201 × 7229231 × 7843701
R_{3X} = 11 × 3E × 591 × 7231 × 78935EX441 × 523074X3XXE × 3266712021E531E1
R_{3E} = 832966217X8X111 × 16EE6202E02X5311278504010EX13001
R_{40} = 5 × 7 × 11 × 15 × 17 × 25 × 75 × 81 × 111 × 141 × 175 × 4541 × EE01 × 1E601 × 106X95 × 8E5281 × 146609481
In fact, the repunit R_{n} = Π_{d}_{|n, d>1}(Φ_{d}(10)), where Φ_{d}(10) is the dth cyclotomic polynomial evaluated at 10.
Properties Edit
- Any positive multiple of the repunit R_{n} 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 R_{1}^{(b)},...,R_{n}^{(b)}. Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits R_{i}^{(b)} and R_{j}^{(b)} with 1≤i<j≤n such that R_{i}^{(b)} and R_{j}^{(b)} have the same residue modulo n. It follows that R_{j}^{(b)} - R_{i}^{(b)} has residue 0 modulo n, i.e. is divisible by n. R_{j}^{(b)} - R_{i}^{(b)} consists of j - i ones followed by i zeroes. Thus, R_{j}^{(b)} - R_{i}^{(b)} = R_{j}_{-i}^{(b)} x b^{i} . 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 R_{j}_{-i}^{(b)}.
- The Feit–Thompson conjecture is that R_{q}^{(p)} never divides R_{p}^{(q)} for two distinct primes p and q.
- Using the Euclidean Algorithm for repunits definition: R_{1}^{(b)} = 1; R_{n}^{(b)} = R_{n}_{-1}^{(b)} x b + 1, any consecutive repunits R_{n}_{-1}^{(b)} and R_{n}^{(b)} are relatively prime in any base b for any n.
- If m and n are relatively prime, R_{m}^{(b)} and R_{n}^{(b)} are relatively prime in any base b for any m and n. The Euclidean Algorithm is based on gcd(m, n) = gcd(m - n, n) for m > n. Similarly, using R_{m}^{(b)} - R_{n}^{(b)} × b^{m}^{-n} = R_{m}_{-n}^{(b)}, it can be easily shown that gcd(R_{m}^{(b)}, R_{n}^{(b)}) = gcd(R_{m}_{-n}^{(b)}, R_{n}^{(b)}) for m > n. Therefore if gcd(m, n) = 1, then gcd(R_{m}^{(b)}, R_{n}^{(b)}) = R_{1}^{(b)} = 1.
- The remainder of R_{n}^{(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 (=11_{4}).
- The only one repunit prime in base 8 is 61 (=111_{8}).
- The only one repunit prime in base 14 is 15 (=11_{14}).
- The only one repunit prime in base 23 is 531 (=111_{23}).
- The only one repunit prime in base 30 is 31 (=11_{30}).
- The only one repunit prime in base 84 is 85 (=11_{84}).
- The only one repunit prime in base X8 is 5E70EX5X8801 (=1111111_{X8}).
- There are no repunit primes in base 9, 21, 28, 41, 54, 69, X1, X5, 100, ..., because of algebraic factors. Especially, all repunits in base 9 are triangular numbers.
- Every perfect power base has at most one generalized repunit prime (since generalized repunits in these bases can be factored algebraically), and it is conjectured every 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 proven repunit prime with length < 1000, besides, base 43 and base 77 also have one known repunit PRP with length < 3000, (43^2545-1)/42 and (77^2685-1)/76.
List of repunit primes base b Edit
We also consider “negative primes” (e.g. R_{2}(−10) = −E) as primes, since they are primes in the domain Z (the set of all integers).
b | length of repunit primes in base b (up to 1000) |
---|---|
−40 | 2, 5, 15, XE |
−3E | 5, 17, 1E, 67 |
−3X | 7, 1E, 4E, 5E, 8E, 167, 237 |
−39 | 87, 111 |
−38 | 2, 7 |
−37 | 5, 7, 17, 18E, 1E1, 27E, 35E |
−36 | 2, 3, 4E1, E45 |
−35 | 15, 497 |
−34 | 45, 57, 855 |
−33 | 3, 11, 105 |
−32 | 2, 5, 11E, 747, E11 |
−31 | 5, 7 |
−30 | 27, 13E, 195, 267 |
−2E | E, 11, 67, X7, 35E, 435, 4E1, 5E5, X4E |
−2X | 3 |
−29 | 5, 57, 111 |
−28 | 2 (no others) |
−27 | 91, 325, 745 |
−26 | 2, E7, 125, 397, 591 |
−25 | 7 |
−24 | 3, 17, 271, 2XE, 34E, 71E |
−23 | (none) |
−22 | E, 91, 16E, 1E1, 24E, 5E5 |
−21 | 3, 7, 1E, 25, 4E, 881, EX5 |
−20 | 2, 7, E, 17 |
−1E | E, 11, 57, 91, 237, 40E |
−1X | 3, 5, 11, 37, 67, 85, 8E, 16E, 255 |
−19 | 3, 5, 7, 11, 31, 24E |
−18 | 2, 5, 67, 75, 4E1, 565, 80E |
−17 | 15, 31, 111, 117, 447 |
−16 | 2, 3, 7, 1E, 61, 511, 665, 775 |
−15 | 7, 15, 1E, 3E, 687 |
−14 | 3, 5, 7, 1E, 31, 75, 105, 125, 18E, 217, 225 |
−13 | 3, 7, 25, 76E |
−12 | 2, 7, 45, 35E, 865 |
−11 | 3, E, 15, 17, 647, 7EE |
−10 | 2, 5, E, 91, 141, X37 |
−E | 5, 7, 12E, 171, 307, 3X5 |
−X | 5, 7, 17, 27, 45, 57, 205, 455 |
−9 | 3, 4E, 167, 397, 545, 701 |
−8 | 2 (no others) |
−7 | 3, 15, 1E, 25, 3E, 51, E2E |
−6 | 2, 3, E, 27, 37, 3E, 4E, 8E, 577 |
−5 | 5, 57, 85, 87, 171, 24E |
−4 | 2, 3 (no others) |
−3 | 2, 3, 5, 7, 11, 1E, 37, 1E5, 25E, 347, 401, XE7, E67 |
−2 | 3, 4, 5, 7, E, 11, 15, 17, 1E, 27, 37, 51, 67, 85, X7, 11E, 13E, 147, 221, 24E, 4X5, EX5 |
2 | 2, 3, 5, 7, 11, 15, 17, 27, 51, 75, 8E, X7, 375, 427, 8X7 |
3 | 3, 7, 11, 5E, 87, 391, 76E, 95E, E37 |
4 | 2 (no others) |
5 | 3, 7, E, 11, 3E, X7, 105, 131, 437, 655 |
6 | 2, 3, 7, 25, 5E, X7, 1X7, 365, 735 |
7 | 5, 11, XE, 105, E97 |
8 | 3 (no others) |
9 | (none) |
X | 2, 17, 1E, 225, 71E |
E | 15, 17, 61, E7, 637 |
10 | 2, 3, 5, 17, 81, 91, 225, 255, 4X5 |
11 | 5, 7, E5, 1E7, 617, 6X7, 711, 835 |
12 | 3, 7, 17, 27, 35 |
13 | 3, 37, 61, 347 |
14 | 2 (no others) |
15 | 3, 5, 7, E, 3E, 5E, 2XE |
16 | 2 |
17 | 17, 27, 3E, 4E, 51, 8E, 241, 745 |
18 | 3, E, 15, X3E |
19 | 3, E, 15, 37, 1X7 |
1X | 2, 5, 67, 85, 25E, 5E5 |
1E | 5 |
20 | 3, 5, 17, 45, 5E, 465, 471 |
21 | (none) |
22 | 7, 37, 24E |
23 | 3 (no others) |
24 | 2, 5, 15, 321, 9X7 |
25 | 5, 107 |
26 | 2, 5, E, 117, 3E5 |
27 | 7, 15, 27 |
28 | (none) |
29 | 3, 145 |
2X | 11, X45 |
2E | 221, 901 |
30 | 2 (no others) |
31 | 11, 5E, 131, 18E, 327, 375 |
32 | 3, 7, 295, 315 |
33 | 251, 447 |
34 | 2, 5, 7, 17, 1E, 25, 391, 527, 8X5 |
35 | 3, 6E, 1X5, 2X1 |
36 | 2, 91E |
37 | 5, 11 |
38 | 5, 27, 11E |
39 | 17, 45, 11E |
3X | 2, 7, 17, 57, 157, 301 |
3E | X7 |
40 | 17, 1X5, 251, 27E, 907 |