The **largest known prime** as of December 2018 is \(2^{82,589,933} − 1\), which has 24,862,048 digits.^{[1]} A prime number is an integer greater than 1 that has no divisors other than 1 and itself. It is well known that there are infinitely many prime numbers (as proven by Euclid), so the search for very large prime numbers is limitless. The Electronic Frontier Foundation gives monetary prizes to people who discover new large primes.^{[2]}

## Records

The most efficient known algorithm for finding large prime numbers is the Lucas-Lehmer test, which tests Mersenne primes. Thus the largest known primes have been Mersenne primes for a long time. George Woltman's distributed computing program GIMPS, an implementation of the Lucas-Lehmer test, has found all the new records since 1996.

The last time a non-Mersenne prime was the largest known prime was in 1992.

A list of record primes (as of January 2020) is given below^{[3]}:

Rank | Form | Prime number | Year found | Number of digits |
---|---|---|---|---|

1 | Mersenne 51** | \(2^{82,589,933}−1\) | 2018 | 24,862,048 |

2 | Mersenne 50** | \(2^{77,232,917}−1\) | 2017 | 23,249,425 |

3 | Mersenne 49** | \(2^{74,207,281}−1\) | 2016 | 22,338,618 |

4 | Mersenne 48* | \(2^{57,885,161}−1\) | 2013 | 17,425,170 |

5 | Mersenne 47 | \(2^{43,112,609}−1\) | 2008 | 12,978,189 |

6 | Mersenne 46 | \(2^{42,643,801}−1\) | 2009 | 12,837,064 |

7 | Mersenne 45 | \(2^{37,156,667}−1\) | 2008 | 11,185,272 |

8 | Mersenne 44 | \(2^{32,582,657}−1\) | 2006 | 9,808,358 |

9 | Non-Mersenne^{[4]}
| \(10\,223 \cdot 2^{31,172,165}+1\) | 2016 | 9,383,761 |

10 | Mersenne 43 | \(2^{30,402,457}−1\) | 2005 | 9,152,052 |

11 | Mersenne 42 | \(2^{25,964,951}−1\) | 2005 | 7,816,230 |

12 | Mersenne 41 | \(2^{24,036,583}−1\) | 2004 | 7,235,733 |

13 | Mersenne 40 | \(2^{20,996,011}−1\) | 2003 | 6,320,430 |

14 | Generalized Fermat prime | \(1\,059\,094^{1\,048\,576}+1\)^{[5]}
| 2018 | 6,317,602 |

15 | Generalized Fermat prime | \(919\,444^{1\,048\,576}+1\)^{[6]}
| 2017 | 6,253,210 |

16 | Sierpinski prime | \(168\,451\cdot2^{19\,375\,200}+1\) | 2017 | 5,832,522 |

17 | Non-Mersenne | \(123\,447^{1\,048\,576}-123\,447^{524\,288}+1\) | 2017 | 5,338,805 |

18 | Non-Mersenne | \(7\cdot6^{6\,772\,401}+1\) | 2019 | 5,269,954 |

19 | Woodall prime (Largest known Woodall prime) | \(8\,508\,301\cdot2^{17\,016\,603}-1\) | 2018 | 5,122,515 |

20 | Non-Mersenne | \(99\,739\cdot2^{14\,019\,102}+1\) | 2019 | 4,220,176 |

21 | Generalized Woodall | \(2\,740\,879\cdot2^{13\,704\,395}-1 = 2\,740\,879 \cdot 32^{2\,740\,879} - 1\) | 2019 | 4,125,441 |

22 | Generalized Woodall | \(479\,216\cdot3^{8\,625\,889}-1 = 1\,437\,648 \cdot 729^{1\,437\,648} - 1\) | 2019 | 4,115,601 |

23^{[7]}
| Non-Mersenne | \(143\,332^{786\,432}-143\,332^{393\,216}+1\) | 2017 | 4,055,114 |

24 | Mersenne 39 | \(2^{13,466,917}−1\) | 2001 | 4,053,946 |

25 | Generalized Cullen prime | \(2\,805\,222\cdot5^{5\,610\,444}+1\ = 2\,805\,222\cdot25^{2\,805\,222}+1\) | 2019 | 3,921,539 |

60 | Mersenne 38 | \(2^{6,972,593}−1\) | 1999 | 2,098,960 |

644 | Mersenne 37 | \(2^{3,021,377}−1\) | 1998 | 909,526 |

662 | Mersenne 36 | \(2^{2,976,221}−1\) | 1997 | 895,932 |

5,381^{[8]}
| Mersenne 35 | \(2^{1,398,269}−1\) | 1996 | 420,921 |

## Proof of the infinitude of primes

Euclid gives an elegant proof that there are infinite prime numbers.

Suppose there is a finite number of prime numbers *p*_{1}*p*_{2}*p*_{3}...*p*_{n}, and let their product be *P*. Then *P* + 1 is one more than a multiple of *p*_{1}, and one more than a multiple of *p*_{2}, etc. *P* + 1 is not divisible by any of our primes, and thus it has no prime factors. Since *P* + 1 > 1, this is impossible.

## Sources

- ↑ The Largest Known Primes
- ↑ EFF Cooperative Computing Awards
- ↑ The Top Twenty: Largest Known Primes. Retrieved 2018-04-13.
- ↑ The largest known primes. Retrieved 2017-11-04.
- ↑ Press release about discovery of 919,444
^{1,048,576}+1. Retrieved 2018-12-21. - ↑ Press release about discovery of 919,444
^{1,048,576}+1. Retrieved 2017-11-04. - ↑ The Prime Database: Phi(3, -143,332^393,216). Retrieved 2018-04-07.
- ↑ The Prime Database: 2^1398269-1. Retrieved 2019-09-14.