A prime number is an integer greater than 1 that has no divisors other than 1 and itself. The largest known prime as of December 2018 is \(2^{82,589,933} − 1\), which has 24,862,048 digits.[1]. 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 December 2023) is given below[3][4]:
Rank | Form | Prime number | Year found | Number of digits |
---|---|---|---|---|
1 | Mersenne 51? [5] | \(2^{82,589,933}−1\) | 2018 | 24,862,048 |
2 | Mersenne 50? [5] | \(2^{77,232,917}−1\) | 2017 | 23,249,425 |
3 | Mersenne 49? [5] | \(2^{74,207,281}−1\) | 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 | Non-Mersenne[6] | \(\varphi(3,-516693^{1048576})\) | 2023 | 11,981,518 |
8 | Non-Mersenne[7] | \(\varphi(3,-465859^{1048576})\) | 2023 | 11,887,192 |
9 | Mersenne 45 | \(2^{37,156,667}−1\) | 2008 | 11,185,272 |
10 | Mersenne 44 | \(2^{32,582,657}−1\) | 2006 | 9,808,358 |
11 | Non-Mersenne[4] | \(10\,223 \cdot 2^{31,172,165}+1\) | 2016 | 9,383,761 |
12 | Mersenne 43 | \(2^{30,402,457}−1\) | 2005 | 9,152,052 |
13 | Mersenne 42 | \(2^{25,964,951}−1\) | 2005 | 7,816,230 |
14 | Mersenne 41 | \(2^{24,036,583}−1\) | 2004 | 7,235,733 |
15 | Generalized Fermat prime | \(1\,963\,736^{2^{20}}+1\) | 2022 | 6,598,776 |
16 | Generalized Fermat prime | \(1\,951\,734^{2^{20}}+1\) | 2022 | 6,595,985 |
17 | Sierpinski prime | \(202\,705\cdot2^{21\,320\,516}+1\) | 2021 | 6,418,121 |
18 | Mersenne 40 | \(2^{20,996,011}−1\) | 2003 | 6,320,430 |
19 | Generalized Fermat prime | \(1\,059\,094^{2^{20}}+1\)[8] | 2018 | 6,317,602 |
20 | Thabit prime[9] | \(3\times 2^{20,928,756}-1\) | 2023 | 6,300,184 |
21 | Generalized Fermat prime | \(919\,444^{2^{20}}+1\)[10] | 2017 | 6,253,210 |
22 | Non-Mersenne (Proth) | \(81\cdot 2^{20498148}+1\) | 2023 | 6,170,560 |
23 | Non-Mersenne (Proth) | \(7\cdot 2^{20267500}+1\) | 2022 | 6,101,127 |
24 | Sierpinski prime | \(168\,451\cdot2^{19\,375\,200}+1\) | 2017 | 5,832,522 |
25 | Riesel prime[11] | \(69\cdot 2^{19,374,980}-1\) | 2022 | 5,832,452 |
Mersenne 39 | \(2^{13,466,917}−1\) | 2001 | 4,053,946 | |
Mersenne 38 | \(2^{6,972,593}−1\) | 1999 | 2,098,960 | |
Mersenne 37 | \(2^{3,021,377}−1\) | 1998 | 909,526 | |
Mersenne 36 | \(2^{2,976,221}−1\) | 1997 | 895,932 | |
Mersenne 35 | \(2^{1,398,269}−1\) | 1996 | 420,921 |
Note that \(\varphi\) is a cyclotomic polynomial, where \(\varphi(3,x)=x^2+x+1\).
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 p1p2p3...pn, and let their product be P. Then P + 1 is one more than a multiple of p1, and one more than a multiple of p2, 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[]
- ↑ Caldwell, Chris. The Largest Known Primes. Retrieved 2023-03-11.
- ↑ EFF Cooperative Computing Awards
- ↑ The Top Twenty: Largest Known Primes. Retrieved 2023-12-26.
- ↑ 4.0 4.1 https://t5k.org/primes/lists/all.txt. Retrieved 2023-12-26.
- ↑ 5.0 5.1 5.2 Not all Mersenne primes tested in this range
- ↑ PrimePage Primes: Phi(3, - 516693^1048576). Retrieved 2023-12-26.
- ↑ PrimePage Primes: Phi(3, - 465859^1048576). Retrieved 2023-06-05.
- ↑ Press release about discovery of 919,4441,048,576+1. Retrieved 2018-12-21.
- ↑ PrimePage Primes: 3 · 2^20928756 - 1. Retrieved 2023-07-17.
- ↑ Press release about discovery of 919,4441,048,576+1. Retrieved 2017-11-04.
- ↑ PrimePage Primes: 69*2^19374980-1. Retrieved 2022-07-22.
External links[]
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