repunit

The term repunit comes from the words 'repeated' and 'unit;' so repunits are positive integers in which every digit is one. (This term was coined by A. H. Beiler in [Beiler1964].) For example, R₁=1, R₂=11, R₃=111, and R_n=(10ⁿ-1)/9. Notice R_n divides R_m whenever n divides m.

Repunit primes are repunits that are prime. For example, 11, 1111111111111111111, and 11111111111111111111111 (2, 19, and 23 digits). The only other known repunit primes are the ones with 317 digits: (10³¹⁷-1)/9; and with 1,031 digits: (10¹⁰³¹-1)/9.

During 1999 Dubner discovered R₄₉₀₈₁ = (10⁴⁹⁰⁸¹-1)/9 was a probable prime, and in October 2000, Lew Baxter discovered the next repunit probable prime after that is R₈₆₄₅₃. In 2007 the probable primes R₁₀₉₂₉₇ (Bourdelais and Dubner) and R₂₇₀₃₄₃ (Voznyy and Budnyy) were found. In 2021 the probable primes R_5794777 and R_8177207 were found (Batalov and Propper). It will be quite some time before these giants are proven prime! As the poet wrote:

Ah, but a man's reach should exceed his grasp, or what's a heaven for? (Robert Browning)

Even though only a few are known, it has been conjectured that there are infinitely many repunit primes. To see why just look at the graph of the known repunit primes and probable primes (here we graph log(log(R_n)) versus n).

Because of their very special form, a repunit prime is also a circular prime and a palindromic prime.

See Also: GeneralizedRepunit

References:

Beiler1964

A. Beiler, Recreations in the theory of numbers, Dover Pub., New York, NY, 1964.

BLSTW88

J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of bⁿ ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., 1988.  Providence RI, pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)

Dubner2002

Dubner, Harvey, "Repunit R₄₉₀₈₁ is a probable prime," Math. Comp., 71:238 (2002) 833--835 (electronic).  (http://dx.doi.org/10.1090/S0025-5718-01-01319-9) MR 1885632 (Abstract available)

WD86

H. C. Williams and H. Dubner, "The primality of R1031," Math. Comp., 47:176 (1986) 703--711.  MR 87k:11141

Yates82

S. Yates, Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982.  pp. vi+215, MR 83k:10014