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 primes and probable primes 
occur for 
,
19, 23, 317, 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 | proven prime (PrimePages 2023, Enge 2024) |
| 109297 | P. Bourdelais (2007), H. Dubner (2007) | Mar. 26-28, 2007 | proven prime (PrimePages 2025, Enge 2026) |
| 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
base-10 repunit 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). A. Enge proved 
to be prime in May
2023, and Enge and P. Underwood proved 
to be prime in May
2025 using elliptic curve primality
proving (PrimePages 2023, PrimePages 2025, Enge 2026). The latter is the largest
proven repunit prime.
Every base-10 repunit prime is a circular prime.
The least positive integer values of 
such that 
is prime for 
, 3, ... are 2, 3, 2, 3, 2, 5, 3, 0,
2, 17, 2, 5, ... (OEIS A084740), and the least
values of 
such that 
is prime for 
, 3, ... are 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, ... (OEIS
A084742). In both sequences,
0 indicates that no such exponent exists.
Williams and Seah (1979) tabulated prime and probable prime generalized repunits for 
and 
. A (base-10) repunit can be prime
only if 
is prime, since for 
with integers 
, 
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 odd primes.
 | OEIS |  of prime or probable-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 a complete prime factorization of any base-10 repunit (McDaniel 1987, pp. 76-78).
is a Probable Prime." Math. Comput. 71, 833-835, 2002. 
." J. Int. Sequences 3, No. 00.2.7,
2000. 
." §A3 in 
." 
-Smith Numbers." Fib. Quart. 25, 76-80,
1987. 
."
May 15, 2023. 
."
May 26, 2025. 
." 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.