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Primes

A spiral diagram showing numbers from 1 to 1199, which numbers are and aren't prime.

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[]

  1. Caldwell, Chris. The Largest Known Primes. Retrieved 2023-03-11.
  2. EFF Cooperative Computing Awards
  3. The Top Twenty: Largest Known Primes. Retrieved 2023-12-26.
  4. 4.0 4.1 https://t5k.org/primes/lists/all.txt. Retrieved 2023-12-26.
  5. 5.0 5.1 5.2 Not all Mersenne primes tested in this range
  6. PrimePage Primes: Phi(3, - 516693^1048576). Retrieved 2023-12-26.
  7. PrimePage Primes: Phi(3, - 465859^1048576). Retrieved 2023-06-05.
  8. Press release about discovery of 919,4441,048,576+1. Retrieved 2018-12-21.
  9. PrimePage Primes: 3 · 2^20928756 - 1. Retrieved 2023-07-17.
  10. Press release about discovery of 919,4441,048,576+1. Retrieved 2017-11-04.
  11. PrimePage Primes: 69*2^19374980-1. Retrieved 2022-07-22.

External links[]

Dynamic googolisms

Involves Dynamic Variables: Lynz · Clarkkkkson · Clarkkkksonplex · C.-R. Number · Eccentric Erbillion · Dimenday · Daygathor · Popacthulhum · Shiki no kazu · -
Largest known...: Alternating Factorial Prime · Carol Prime · Prime

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