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.