Repunit

A (generalized) repunit to the base $b$ is a number of the form

$$M^b_n = {b^n-1\over b-1}.$$

The term ``repunit'' was coined by Beiler (1966), who also gave the first tabulation of known factors. Repunits $M_n=M_n^2=2^n-1$ with $b=2$ are called Mersenne Numbers. If $b=10$, the number is called a repunit (since the digits are all 1s). A number of the form

$$R_n={10^n-1\over 10-1}=R_n={10^n-1\over 9}$$

is therefore a (decimal) repunit of order $n$.

$b$	Sloane	$b$-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 $3\leq b\leq 12$ and $2\leq n\leq 1000$. A (base-10) repunit can be Prime only if $n$ is Prime, since otherwise $10^{ab}-1$ is a Binomial Number which can be factored algebraically. In fact, if $n=2a$ is Even, then $10^{2a}-1=(10^a-1)(10^a+1)$. The only base-10 repunit Primes $R_n$ for $n\leq 16,500$ are $n=2$, 19, 23, 317, and 1031 (Sloane's A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987). The number of factors for the base-10 repunits for $n=1$, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (Sloane's A046053).

$b$	Sloane	$n$ of Prime $b$-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 Brillhart et al. 1988) is maintained on-line. These tables include factors for $10^n-1$ (with $n\leq 209$ odd) and $10^n+1$ (for $n\leq 210$ Even and Odd) in the files ftp://sable.ox.ac.uk/pub/math/cunningham/10- and ftp://sable.ox.ac.uk/pub/math/cunningham/10+. After algebraically factoring $R_n$, these are sufficient for complete factorizations. Yates (1982) published all the repunit factors for $n\leq 1000$, a portion of which are reproduced in the Mathematica ${}^{\scriptstyle\circledRsymbol}$ 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 $b^n\pm 1$, $b=2$, $3, 5, 6, 7, 10, 11, 12$ 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.

Dubner, H. ``Generalized Repunit Primes.'' Math. Comput. 61, 927-930, 1993.

Guy, R. K. ``Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape $k\cdot 2^n+2$.'' §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.

Snyder, W. M. ``Factoring Repunits.'' Am. Math. Monthly 89, 462-466, 1982.

Weisstein, E. W. ``Repunits.'' Mathematica notebook Repunit.m.

Williams, H. C. and Dubner, H. ``The Primality of $R1031$.'' Math. Comput. 47, 703-711, 1986.

Williams, H. C. and Seah, E. ``Some Primes of the Form $(a^n-1)/(a-1)$. Math. Comput. 33, 1337-1342, 1979.

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.