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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

R_n=(10^n-1)/(10-1)
(1)
=(10^n-1)/9.
(2)

Repunits R_n therefore have exactly n decimal digits. Amazingly, the squares of the repunits R_n^2 give the Demlo numbers, 1^2=1, 11^2=121, 111^2=12321, ... (OEIS A002275 and A002477).

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, ... (OEIS A046053).

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

Repunits can be generalized to base b, giving a base-b repunit as number of the form

 M_n^((b))=(b^n-1)/(b-1).
(3)

This gives the special cases summarized in the following table.

bM_n^((b))name
22^n-1Mersenne number M_n
10(10^n-1)/9repunit R_n

The idea of repunits can also be extended to negative bases. Except for requiring n to be odd, the math is very similar (Dubner and Granlund 2000).

bOEISb-repunits
-3A0664431, 7, 61, 547, 4921, 44287, 398581, ...
-2A0075831, 3, 11, 43, 171, 683, 2731, ...
2A0002251, 3, 7, 15, 31, 63, 127, ...
3A0034621, 4, 13, 40, 121, 364, ...
4A0024501, 5, 21, 85, 341, 1365, ...
5A0034631, 6, 31, 156, 781, 3906, ...
6A0034641, 7, 43, 259, 1555, 9331, ...
7A0230001, 8, 57, 400, 2801, 19608, ...
8A0230011, 9, 73, 585, 4681, 37449, ...
9A0024521, 10, 91, 820, 7381, 66430, ...
10A0022751, 11, 111, 1111, 11111, ...
11A0161231, 12, 133, 1464, 16105, 177156, ...
12A0161251, 13, 157, 1885, 22621, 271453, ...

See also

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.

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Repunit

Cite this as:

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

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