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# 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

 (1) (2)

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).

A repunit that is a prime number is known as a repunit prime.

Repunits 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, ...

Cunningham Number, Demlo Number, Fermat Number, Mersenne Number, Repdigit, Repunit Prime, Smith Number

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## References

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.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." http://www.swox.com/gmp/repunit.html.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, A004061/M2620, A004062/M0861, A004063/M3836, A004064/M0744, A005808/M5032, A016123, A016125, A023000, A023001, A028491/M2643, A046053, A057171, A057172, A057173, A057175, A057177, A057178, A066443, and A084740 in "The On-Line Encyclopedia of Integer Sequences."Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146, 1969.Yates, S. "The Mystique of Repunits." Math. Mag. 51, 22-28, 1978.Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates, 1982.

Repunit

## Cite this as:

Weisstein, Eric W. "Repunit." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Repunit.html