Repunit
A (generalized) repunit to the base 
is a number of the form
The term ``repunit'' was coined by Beiler (1966), who also gave the first tabulation of known factors. Repunits
with 
are called Mersenne Numbers. If 
, the number is called a repunit(since the digits are all 1s). A number of the form
is therefore a (decimal) repunit of order
. | Sloane | -Repunits |
| 2 | Sloane's A000225 | 1, 3, 7, 15, 31, 63, 127, ... |
| 3 | Sloane's A003462 | 1, 4, 13, 40, 121, 364, ... |
| 4 | Sloane's A002450 | 1, 5, 21, 85, 341, 1365, ... |
| 5 | Sloane's A003463 | 1, 6, 31, 156, 781, 3906, ... |
| 6 | Sloane's A003464 | 1, 7, 43, 259, 1555, 9331, ... |
| 7 | Sloane's A023000 | 1, 8, 57, 400, 2801, 19608, ... |
| 8 | Sloane's A023001 | 1, 9, 73, 585, 4681, 37449, ... |
| 9 | Sloane's A002452 | 1, 10, 91, 820, 7381, 66430, ... |
| 10 | Sloane's A002275 | 1, 11, 111, 1111, 11111, ... |
| 11 | Sloane's A016123 | 1, 12, 133, 1464, 16105, 177156, ... |
| 12 | Sloane's A016125 | 1, 13, 157, 1885, 22621, 271453, ... |
Williams and Seah (1979) factored generalized repunits for 
and 
. A (base-10) repunit can bePrime only if is Prime, since otherwise 
is a Binomial Number which can be factored algebraically.In fact, if 
is Even, then 
. The only base-10 repunit Primes 
for 
are
, 19, 23, 317, and 1031 (Sloane's A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987). The number of factorsfor the base-10 repunits for 
, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (Sloane's A046053).
| Sloane | of Prime -Repunits |
| 2 | Sloane's A000043 | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, ... |
| 3 | Sloane's A028491 | 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, ... |
| 5 | Sloane's A004061 | 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, ... |
| 6 | Sloane's A004062 | 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, ... |
| 7 | Sloane's A004063 | 5, 13, 131, 149, 1699, ... |
| 10 | Sloane's A004023 | 2, 19, 23, 317, 1031, ... |
| 11 | Sloane's A005808 | 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, ... |
| 12 | Sloane's A004064 | 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, ... |
A table of the factors not obtainable algebraically for generalized repunits (a continuously updated version of Brillhartet al. 1988) is maintained on-line. These tables include factors for 
(with 
odd) and 
(for 
ftp://sable.ox.ac.uk/pub/math/cunningham/10- andftp://sable.ox.ac.uk/pub/math/cunningham/10+. After algebraically factoring , these are sufficient for completefactorizations. Yates (1982) published all the repunit factors for 
, a portion of which are reproduced in theMathematica
notebook by Weisstein.
A Smith Number can be constructed from every factored repunit.
See also Cunningham Number, Fermat Number, Mersenne Number, Repdigit, Smith Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 66, 1987. 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 Dubner, H. ``Generalized Repunit Primes.'' Math. Comput. 61, 927-930, 1993. Guy, R. K. ``Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape 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. Snyder, W. M. ``Factoring Repunits.'' Am. Math. Monthly 89, 462-466, 1982. Williams, H. C. and Dubner, H. ``The Primality of Williams, H. C. and Seah, E. ``Some Primes of the Form 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.
, , 
Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988. Updates are available electronically from ftp://sable.ox.ac.uk/pub/math/cunningham.
.'' §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.
Weisstein, E. W. ``Repunits.'' Mathematica notebook Repunit.m.
.'' Math. Comput. 47, 703-711, 1986.
. Math. Comput. 33, 1337-1342, 1979.
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