# 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 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. 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