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The largest known prime as of January 2016 is $$2^{74,207,281} − 1$$, which has 22,338,618 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.

A list of record primes (as of January 19, 2016) is given below:

Rank Form Prime number Year found Number of digits
1 Mersenne 49 $$2^{74,207,281}−1$$ 2016 22338618
2 Mersenne 48 $$2^{57,885,161}−1$$ 2013 17425170
3 Mersenne 47 $$2^{43,112,609}−1$$ 2008 12978189
4 Mersenne 46 $$2^{42,643,801}−1$$ 2009 12837064
5 Mersenne 45 $$2^{37,156,667}−1$$ 2008 11185272
6 Mersenne 44 $$2^{32,582,657}−1$$ 2006 9808358
7 Mersenne 43 $$2^{30,402,457}−1$$ 2005 9152052
8 Mersenne 42 $$2^{25,964,951}−1$$ 2005 7816230
9 Mersenne 41 $$2^{24,036,583}−1$$ 2004 7235733
10 Mersenne 40 $$2^{20,996,011}−1$$ 2003 6320430
11 Mersenne 39 $$2^{13,466,917}−1$$ 2001 4053946
12 Non-Mersenne[3] $$19249\cdot2^{13,018,586}−1$$ 2007 3918990
15 Non-Mersenne $$3\cdot2^{11,484,018}-1$$ 2014 3457035
16 Non-Mersenne $$3\cdot2^{10,829,346}+1$$ 2014 3259959
17 Non-Mersenne $$475856^{524288}+1$$ 2012 2976633
18 Non-Mersenne $$356926^{524288}+1$$ 2012 2911151
29 Mersenne 38 $$2^{6,972,593}−1$$ 1999 2098960
194 Mersenne 37 $$2^{3,021,377}−1$$ 1998 909526
205 Mersenne 36 $$2^{2,976,221}−1$$ 1997 895932
2134[4] Mersenne 35 $$2^{1,398,269}−1$$ 1996 420921

## 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 largest known primes. Retrieved 2015-01-06.
4. The Prime Database: 2^1398269-1. Retrieved 2014-05-10.