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 be Prime 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 factors for 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 Brillhart
*et al. * 1988) is maintained on-line. These tables include factors for (with odd) and (for 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 , these are sufficient for complete
factorizations. Yates (1982) published all the repunit factors for , a portion of which are reproduced in the
*Mathematica*
notebook by Weisstein.

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.

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 , ,
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 .'' §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 .'' *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. ``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.

© 1996-9

1999-05-25