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The largest known prime as of January 2018 is $$2^{77,232,917} − 1$$, which has 23,249,425 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 Edit

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 April 13, 2018) is given below[3]:

Rank Form Prime number Year found Number of digits
1 Mersenne 50** $$2^{77,232,917}−1$$ 2017 23,249,425
2 Mersenne 49** $$2^{74,207,281}−1$$ 2016 22,338,618
3 Mersenne 48* $$2^{57,885,161}−1$$ 2013 17,425,170
4 Mersenne 47* $$2^{43,112,609}−1$$ 2008 12,978,189
5 Mersenne 46 $$2^{42,643,801}−1$$ 2009 12,837,064
6 Mersenne 45 $$2^{37,156,667}−1$$ 2008 11,185,272
7 Mersenne 44 $$2^{32,582,657}−1$$ 2006 9,808,358
8 Non-Mersenne[4] $$1.0223 \cdot 2^{31,172,165}+1$$ 2016 9,383,761
9 Mersenne 43 $$2^{30,402,457}−1$$ 2005 9,152,052
10 Mersenne 42 $$2^{25,964,951}−1$$ 2005 7,816,230
11 Mersenne 41 $$2^{24,036,583}−1$$ 2004 7,235,733
12 Mersenne 40 $$2^{20,996,011}−1$$ 2003 6,320,430
13 Generalized Fermat prime $$919\,444^{1\,048\,576}+1$$[5] 2017 6,253,210
14 Sierpinski prime $$168\,451*2^{19\,375\,200}+1$$ 2017 5,832,522
15 Non-Mersenne $$123\,447^{1\,048\,576}-123\,447^{524\,288}+1$$ 2017 5,338,805
16 Woodall prime $$8\,508\,301*2^{17\,016\,603}-1$$ 2018 5,122,515
17[6] Non-Mersenne $$143\,332^{786\,432}-143\,332^{393\,216}+1$$ 2017 4,055,114
18 Mersenne 39 $$2^{13,466,917}−1$$ 2001 4,053,946
19 Non-Mersenne[7] $$1.9249\cdot2^{13,018,586}−1$$ 2007 3,918,990
20 Non-Mersenne $$3\cdot2^{11,895,718}-1$$ 2015 3,580,969
21 Non-Mersenne $$3\cdot2^{11,731,850}-1$$ 2015 3,531,640
22 Non-Mersenne $$3\cdot2^{11,484,018}-1$$ 2014 3,457,035
23 Non-Mersenne $$193\,997*2^{11\,452\,891}+1$$ 2018 3,447,670
24 Non-Mersenne $$2,061,748^{524\,288}+1$$ 2018 3,310,478
25 Non-Mersenne $$1,880,370^{524\,288}+1$$ 2018 3,289,511
50 Mersenne 38 $$2^{6,972,593}−1$$ 1999 2,098,960
426 Mersenne 37 $$2^{3,021,377}−1$$ 1998 909,526
444 Mersenne 36 $$2^{2,976,221}−1$$ 1997 895,932
4,034[8] Mersenne 35 $$2^{1,398,269}−1$$ 1996 420,921

## Proof of the infinitude of primes Edit

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 Edit

1. The Largest Known Primes
2. EFF Cooperative Computing Awards
3. The Top Twenty: Largest Known Primes. Retrieved 2018-04-13.
4. The largest known primes. Retrieved 2017-11-04.
5. Press release about discovery of 919,4441,048,576+1. Retrieved 2017-11-04.
6. The Prime Database: Phi(3, -143,332^393,216). Retrieved 2018-04-07.
7. The largest known primes. Retrieved 2015-01-06.
8. The Prime Database: 2^1398269-1. Retrieved 2018-04-07.