A repunit prime is a repunit (i.e., a number consisting of copies of the single digit 1) that is also a prime number.
The base-10 repunit (possibly probable) primes 
occur for 
, 19, 23, 317, and 1031, 49081, 86453, 109297, 270343, ...
(OEIS A004023; Madachy 1979, Williams and Dubner
1986, Ball and Coxeter 1987, Granlund, Dubner 1999, Baxter 2000).
T. Granlund completed a search for probable primes up to 
in 1998 using two months of CPU time on a parallel computer.
The search was extended by Dubner (1999), culminating in the discovery of the probable
prime 
.
A number of larger repunit probable primes have
since been found, as summarized in the following table. As of July 1, 2021, all numbers
up to 
have been searched (OEIS A004023).
 | discoverer(s) | date | status |
| 2 | proven prime | ||
| 19 | proven prime | ||
| 23 | proven prime | ||
| 317 | proven prime | ||
| 1031 | proven prime (Williams and Dubner 1986) | ||
| 49081 | H. Dubner (1999, 2002) | Sep. 9, 1999 | proven prime (Underwood 2022) |
| 86453 | L. Baxter (2000) | Oct. 26, 2000 | probable prime |
| 109297 | P. Bourdelais (2007), H. Dubner (2007) | Mar. 26-28, 2007 | probable prime |
| 270343 | M. Voznyy and A. Budnyy (2007) | Jul. 11, 2007 | probable prime |
| 5794777 | S. Batalov and R. Propper | Apr. 20, 2021 | probable prime |
| 8177207 | S. Batalov and R. Propper | May 8, 2021 | probable prime |
was the largest proven
prime (Williams and Dubner 1986) until 2022, when P. Underwood proved 
to be prime using elliptic
curve primality proving. The certification took 20 months on an AMD 3990x computer
with 64 cores, and verification took about 13 hours (Underwood 2022).
Every prime repunit is a circular prime.
The sequence of least 
such that 
is prime for 
,
2, ... are 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, ... (OEIS A084740),
and the sequence of least 
such that 
is prime for 
, 2, ... are 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, ... (OEIS
A084742).
Williams and Seah (1979) factored generalized repunits for 
and 
. A (base-10) repunit can be prime
only if 
is prime, since otherwise 
is a binomial number
which can be factored algebraically. In fact, if 
is even, then 
. As with positive bases, all the
exponents of prime repunits with negative bases are also prime.
 | OEIS |  of prime  -repunits |
 | A057178 | 5, 11, 109, 193, 1483, ... |
 | A057177 | 5, 7, 179, 229, 439, 557, 6113, ... |
 | A001562 | 5, 7, 19, 31, 53, 67, 293, ... |
 | A057175 | 3, 59, 223, 547, 773, 1009, 1823, ... |
 | A057173 | 3, 17, 23, 29, 47, 61, 1619, ... |
 | A057172 | 3, 11, 31, 43, 47, 59, 107, ... |
 | A057171 | 5, 67, 101, 103, 229, 347, 4013, ... |
 | A007658 | 3, 5, 7, 13, 23, 43, 281, ... |
 | A000978 | 3, 5, 7, 11, 13, 17, 19, ... |
| 2 | A000043 | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, ... |
| 3 | A028491 | 3, 7, 13, 71, 103, 541, 1091, 1367, ... |
| 5 | A004061 | 3, 7, 11, 13, 47, 127, 149, 181, 619, ... |
| 6 | A004062 | 2, 3, 7, 29, 71, 127, 271, 509, 1049, ... |
| 7 | A004063 | 5, 13, 131, 149, 1699, ... |
| 10 | A004023 | 2, 19, 23, 317, 1031, ... |
| 11 | A005808 | 17, 19, 73, 139, 907, 1907, 2029, 4801, ... |
| 12 | A004064 | 2, 3, 5, 19, 97, 109, 317, 353, 701, ... |
Yates (1982) published all the repunit factors for 
. Brillhart et al. (1988) gave a table of repunit
factors which cannot be obtained algebraically, and a continuously updated version
of this table is now maintained online. These tables include factors for 
(with 
odd) and 
(with 
even and odd).
After algebraically factoring 
, these types of factors are sufficient for complete factorizations.
A Smith number can be constructed from every factored repunit.
is a Probable Prime." Math. Comput. 71, 833-835, 2002. 
." J. Int. Sequences 3, No. 00.2.7,
2000. 
." §A3 in 
." Math. Comput. 47, 703-711, 1986.Williams,
H. C. and Seah, E. "Some Primes of the Form 
. Math. Comput. 33, 1337-1342, 1979.Yates,
S. "Prime Divisors of Repunits." J. Recr. Math. 8, 33-38,
1975.Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates,
1982.