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The largest known prime as of December 2018 is $$2^{82,589,933} − 1$$, which has 24,862,048 digits. 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.

## 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 October 2020) is given below:

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 $$10\,223 \cdot 2^{31,172,165}+1$$ 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 $$1\,059\,094^{1\,048\,576}+1$$ 2018 6,317,602
15 Generalized Fermat prime $$919\,444^{1\,048\,576}+1$$ 2017 6,253,210
16 Sierpinski prime $$168\,451\cdot2^{19\,375\,200}+1$$ 2017 5,832,522
17 Proth prime $$7\cdot 2^{18,233,956}+1$$ 2020 5,488,969
18 Non-Mersenne $$123\,447^{1\,048\,576}-123\,447^{524\,288}+1$$ 2017 5,338,805
19 Non-Mersenne $$7\cdot6^{6\,772\,401}+1$$ 2019 5,269,954
20 Woodall prime (Largest known Woodall prime) $$8\,508\,301\cdot2^{17\,016\,603}-1$$ 2018 5,122,515
21 Non-Mersenne (generalized unique) $$2^{15,317,227}+2^{7,658,614}+1$$ 2020 4,610,945
22 Non-Mersenne $$6 \cdot 5^{6\,546\,983} + 1$$ 2020 4,576,146
23 Non-Mersenne $$6\,962 \cdot 31^{2\,863\,120} - 1$$ 2020 4,269,952
24 Non-Mersenne (Sierpinski prime) $$99\,739\cdot2^{14\,019\,102}+1$$ 2019 4,220,176
25 Generalized Woodall $$2\,740\,879\cdot2^{13\,704\,395}-1 = 2\,740\,879 \cdot 32^{2\,740\,879} - 1$$ 2019 4,125,441
28 Mersenne 39 $$2^{13,466,917}−1$$ 2001 4,053,946
84 Mersenne 38 $$2^{6,972,593}−1$$ 1999 2,098,960
895 Mersenne 37 $$2^{3,021,377}−1$$ 1998 909,526
913 Mersenne 36 $$2^{2,976,221}−1$$ 1997 895,932
6,453 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 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.

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