# 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; and with 1,031 digits:
(10^{1031}-1)/9.

During 1999 Dubner discovered R_{49081} = (10^{49081}-1)/9
was a probable prime,
and in October 2000, Lew Baxter discovered the next repunit probable prime after that 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 prime R_{5794777} was found (Batalov and Propper).
It will be some time before these giants is proven prime! As the poet wrote:

Ah, but a mans reach should exceed his grasp, or whats 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})) verses

*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. 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- Yates82
S. Yates,Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982. pp. vi+215,MR 83k:10014