Largest known 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.. It is well known that there are infinitely many prime numbers (as proven by Euclid), so the search for very large s is limitless. The gives monetary prizes to people who discover new large primes.EFF Cooperative Computing Awards
== Approximations in other notations ==
For \(2^{82,589,933} − 1\):
Notation 	Lower bound 	Upper bound
Scientific notation 	\(1.488\times10^{24862047}\) 	\(1.489\times10^{24862047}\)
Up-arrow notation(Fish's definition for non-integers)	\(10 \uparrow\uparrow 2.8\) 	\(10 \uparrow\uparrow 2.9\)
Chained arrow notation 	\(10\rightarrow24862047\) 	\(10\rightarrow24862048\)
Bowers' Exploding Array Function 	{10, 24862047} 	{10, 24862048}
Bashicu matrix system 	(0)(1)[4] 	(0)(1)[5]
Steinhaus-Moser Notation 	7[3][3] 	8[3][3]
Fast-growing hierarchy 	\(f_2(f_2(21))\) 	\(f_2(f_2(22))\)
Hardy hierarchy 	\(H_{\omega^22}(21)\) 	\(H_{\omega^22}(22)\)
Slow-growing hierarchy 	\(g_{\omega^{\omega^7}}(10)\)	\(g_{\omega^{\omega^8}}(10)\)
Hyper-E notation 	E24862047 	E24862048
Hyperfactorial array notation 	(10!)! 	(11!)!
== Records ==
The most efficient known algorithm for finding large prime numbers is the , 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 August 2024) is given below:
Rank	Form 	Prime number 	Year found	Number of digits
1 	Mersenne 51? Not all Mersenne primes tested in this range 	\(2^{82,589,933}−1\) 	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 	Non-MersennePrimePage Primes: Phi(3, - 516693^1048576). Retrieved 2023-12-26.	\(\varphi(3,-516693^{1048576})\) 	2023 	11,981,518
8 	Non-MersennePrimePage Primes: Phi(3, - 465859^1048576). Retrieved 2023-06-05.	\(\varphi(3,-465859^{1048576})\) 	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 (Proth prime) 	\(10\,223 \cdot 2^{31,172,165}+1\) 	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 	Riesel prime 	\(69 \times 2^{24,612,729} - 1\) 	2024 	7,409,172
15 	Mersenne 41 	\(2^{24,036,583}−1\) 	2004 	7,235,733
16 	Riesel prime 	\(107,347 \times 2^{23,427,517} - 1\)	2024 	7,052,391
17 	 	\(1\,963\,736^{2^{20}}+1\) 	2022 	6,598,776
18 	 	\(1\,951\,734^{2^{20}}+1\) 	2022 	6,595,985
19 	 	\(202\,705\cdot2^{21\,320\,516}+1\) 	2021 	6,418,121
20 	Mersenne 40 	\(2^{20,996,011}−1\) 	2003 	6,320,430
21 	 	\(1\,059\,094^{2^{20}}+1\) 	2018 	6,317,602
22 	Thabit primePrimePage Primes: 3 · 2^20928756 - 1. Retrieved 2023-07-17. 	\(3\times 2^{20,928,756}-1\) 	2023 	6,300,184
23 	 	\(919\,444^{2^{20}}+1\) 	2017 	6,253,210
24 	Non-Mersenne (Proth) 	\(81\cdot 2^{20498148}+1\) 	2023 	6,170,560
25 	Non-Mersenne (Proth) 	\(7\cdot 2^{20267500}+1\) 	2022 	6,101,127
 	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 , 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 ..., and let their product be P. Then P + 1 is one more than a multiple of , and one more than a multiple of , 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 ==
== External links ==
*The Largest Known Primes
*5000 largest known primes list
ja:知られている最大の素数
ru:Наибольшее простое число
Category:Numbers
Category:Class 3
Category:Prime numbers
Category:Numerical records
Category:Lower class 3
Category:Odd numbers
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