## 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, R_{1}=1,
R_{2}=11, R_{3}=111, and
R_{n}=(10^{n}-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:
(10^{317}-1)/9, 1,031 digits and
(10^{1031}-1)/9.

During 1999 Dubner discovered
R_{49081} = (10^{49081}-1)/9 was a
probable prime. In October 2000, Lew Baxter
discovered the next repunit probable prime is
R_{86453}. In 2007 the probable primes R_{109297} (Bourdelais and Dubner) and R_{270343} (Voznyy and Budnyy) were found. In 2021 the probable primes R_{5794777} and R_{8177207} were found (Batalov and Propper). It will be some time
before these giant primes are proven! As the poet wrote:

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

What makes these repunits R so difficult to prove prime is that we do not have an easy way to factor R-1 or R+1. The (practically) quick methods of primality proving all require factoring.

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})) verses *n*.

rank prime digits who when comment 1 R(49081)49081 c70 Mar 2022 Repunit, unique, ECPP 2 R(1031)1031 WD Dec 1985 Repunit

- Giovanni Di Maria's Repunit website

- 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. TuckermanandS. S. Wagstaff, Jr.,Factorizations of, Amer. Math. Soc., 1988. Providence RI, pp. xcvi+236, ISBN 0-8218-5078-4.b^{n}± 1,b=2,3,5,6,7,10,12 up to high powersMR 90d:11009(Annotation available)- Dubner2002
Dubner, Harvey, "RepunitR_{49081}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. WilliamsandH. Dubner, "The primality ofR1031,"Math. Comp.,47:176 (1986) 703--711.MR 87k:11141- Williams78b
H. C. Williams, "Some primes with interesting digit patterns,"Math. Comp.,32(1978) 1306--1310. Corrigendum in39(1982), 759.MR 58:484- Yates82
S. Yates,Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982. pp. vi+215,MR 83k:10014

Chris K. Caldwell
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