Repunit
A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.
In base-10, repunits have the form
Repunits 
therefore have exactly 
decimal digits. Amazingly, the squares of the repunits 
give the Demlo numbers, 
, 
, 
, ... (OEIS A002275 and A002477).
The number of factors for the base-10 repunits for 
, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS A046053). The base-10 repunit probable primes 
occur for 
, 19, 23, 317, and 1031, 49081, 86453, 109297, and 270343 (OEIS A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987, Granlund, Dubner 1999, Baxter 2000), where 
is the largest proven prime (Williams and Dubner 1986). T. Granlund completed a search up to 
in 1998 using two months of CPU time on a parallel computer. The search was extended by Dubner (1999), culminating in the discovery of the probable prime 
. A number of larger repunit probable primes have since been found, as summarized in the following table.
 |
discoverer(s) |
date |
| 49081 |
H. Dubner (1999, 2002) |
Sep. 9, 1999 |
| 86453 |
L. Baxter (2000) |
Oct. 26, 2000 |
| 109297 |
P. Bourdelais (2007), H. Dubner (2007) |
Mar. 26-28, 2007 |
| 270343 |
M. Voznyy and A. Budnyy (2007) |
Jul. 11, 2007 |
Every prime repunit is a circular prime.
Repunit can be generalized to base 
, giving a base-
repunit as number of the form
 |
(3)
|
This gives the special cases summarized in the following table.
 |
 |
name |
| 2 |
 |
Mersenne number  |
| 10 |
 |
repunit  |
The idea of repunits can also be extended to negative bases. Except for requiring 
to be odd, the math is very similar (Dubner and Granlund 2000).
 |
OEIS |
 -repunits |
 |
A066443 |
1, 7, 61, 547, 4921, 44287, 398581, ... |
 |
A007583 |
1, 3, 11, 43, 171, 683, 2731, ... |
| 2 |
A000225 |
1, 3, 7, 15, 31, 63, 127, ... |
| 3 |
A003462 |
1, 4, 13, 40, 121, 364, ... |
| 4 |
A002450 |
1, 5, 21, 85, 341, 1365, ... |
| 5 |
A003463 |
1, 6, 31, 156, 781, 3906, ... |
| 6 |
A003464 |
1, 7, 43, 259, 1555, 9331, ... |
| 7 |
A023000 |
1, 8, 57, 400, 2801, 19608, ... |
| 8 |
A023001 |
1, 9, 73, 585, 4681, 37449, ... |
| 9 |
A002452 |
1, 10, 91, 820, 7381, 66430, ... |
| 10 |
A002275 |
1, 11, 111, 1111, 11111, ... |
| 11 |
A016123 |
1, 12, 133, 1464, 16105, 177156, ... |
| 12 |
A016125 |
1, 13, 157, 1885, 22621, 271453, ... |
Williams and Seah (1979) factored generalized repunits for 
and 
. A (base-10) repunit can be prime only if 
is prime, since otherwise 
is a binomial number which can be factored algebraically. In fact, if 
is even, then 
. As with positive bases, all the exponents of prime repunits with negative bases are also prime.
 |
OEIS |
 of prime  -repunits |
 |
A057178 |
5, 11, 109, 193, 1483, ... |
 |
A057177 |
5, 7, 179, 229, 439, 557, 6113, ... |
 |
A001562 |
5, 7, 19, 31, 53, 67, 293, ... |
 |
A057175 |
3, 59, 223, 547, 773, 1009, 1823, ... |
 |
A057173 |
3, 17, 23, 29, 47, 61, 1619, ... |
 |
A057172 |
3, 11, 31, 43, 47, 59, 107, ... |
 |
A057171 |
5, 67, 101, 103, 229, 347, 4013, ... |
 |
A007658 |
3, 5, 7, 13, 23, 43, 281, ... |
 |
A000978 |
3, 5, 7, 11, 13, 17, 19, ... |
| 2 |
A000043 |
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, ... |
| 3 |
A028491 |
3, 7, 13, 71, 103, 541, 1091, 1367, ... |
| 5 |
A004061 |
3, 7, 11, 13, 47, 127, 149, 181, 619, ... |
| 6 |
A004062 |
2, 3, 7, 29, 71, 127, 271, 509, 1049, ... |
| 7 |
A004063 |
5, 13, 131, 149, 1699, ... |
| 10 |
A004023 |
2, 19, 23, 317, 1031, ... |
| 11 |
A005808 |
17, 19, 73, 139, 907, 1907, 2029, 4801, ... |
| 12 |
A004064 |
2, 3, 5, 19, 97, 109, 317, 353, 701, ... |
Yates (1982) published all the repunit factors for 
, a portion of which are reproduced in the Wolfram Language notebook by Weisstein. Brillhart et al. (1988) gave a table of repunit factors which cannot be obtained algebraically, and a continuously updated version of this table is now maintained online. These tables include factors for 
(with 
odd) and 
(with 
even and odd). After algebraically factoring 
, these types of factors are sufficient for complete factorizations.
The sequence of least 
such that 
is prime for 
, 2, ... are 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, ... (OEIS A084740), and the sequence of least 
such that 
is prime for 
, 2, ... are 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, ... (OEIS A084742).
A Smith number can be constructed from every factored repunit.
REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 66, 1987.
Baxter, L. "R86453 Is a New Probable Prime Repunit." 26 Oct 2000. https://listserv.nodak.edu/scripts/wa.exe?A2=ind0010&L=nmbrthry&P=2557.
Beiler, A. H. "11111...111." Ch. 11 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988.
Di Maria, G. "The Repunit Primes Project." https://www.repunit.org/.
Dubner, H. "Generalized Repunit Primes." Math. Comput. 61, 927-930, 1993.
Dubner, H. "New prp Repunit R(49081)." 9 Sep 1999. https://listserv.nodak.edu/scripts/wa.exe?A2=ind9909&L=nmbrthry&P=740.
Dubner, H. "Repunit 
is a Probable Prime." Math. Comput. 71, 833-835, 2002. https://www.ams.org/mcom/2002-71-238/.
Dubner, H. "New Repunit R(109297)." 3 Apr 2007. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178.
Dubner, H. and Granlund, T. "Primes of the Form 
." J. Int. Sequences 3, No. 00.2.7, 2000. https://www.cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html.
Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949.
Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 85-86, 1984.
Granlund, T. "Repunits." https://www.swox.com/gmp/repunit.html.
Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape 
." §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 152-153, 1979.
Ribenboim, P. "Repunits and Similar Numbers." §5.5 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 350-354, 1996.
Sloane, N. J. A. Sequences A000043/M0672, A000225/M2655, A000978, A001562, A002275, A002477/M5386, A002450/M3914, A002452/M4733, A003462/M3463, A007583, A007658, A003463/M4209, A003464/M4425, A004023/M2114, A004023/M2114, A004061/M2620, A004062/M0861, A004063/M3836, A004064/M0744, A005808/M5032, A016123, A016125, A023000, A023001, A028491/M2643, A046053, A057171, A057172, A057173, A057175, A057177, A057178, A066443, A084740, and A084742 in "The On-Line Encyclopedia of Integer Sequences."
Snyder, W. M. "Factoring Repunits." Am. Math. Monthly 89, 462-466, 1982.
Sorli, R. "Factorization Tables." https://www-staff.maths.uts.edu.au/~rons/fact/fact.htm.
Voznyy, M. and Budnyy, A. "New PRP Repunit R(270343)." 15 Jul 2007. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=nmbrthry&T=0&P=1086.
Williams, H. C. and Dubner, H. "The Primality of 
." Math. Comput. 47, 703-711, 1986.
Williams, H. C. and Seah, E. "Some Primes of the Form 
. Math. Comput. 33, 1337-1342, 1979.
Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146, 1969.
Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8, 33-38, 1975.
Yates, S. "The Mystique of Repunits." Math. Mag. 51, 22-28, 1978.
Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates, 1982.
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