Largest known prime number

The largest known prime number is , a number that has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).

thumb|400px|A plot of the number of digits in the largest known prime by year since the advent of the [[electronic computer. The vertical scale is logarithmic.]]

A prime number is a natural number greater than 1 with no divisors other than 1 and itself. Euclid's theorem proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; there is no largest prime.

Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can be verified by a specialized primality test that is faster than the general one. , the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2^k − 1 is simply k ones.

Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not true. While large primes with hundreds of digits are indeed used for cryptography, primes with millions of digits are not.

Current record

thumb| Using standard [[A4 paper format of 50 lines per page and 75 characters per line, it would require 10,940 single-sided pages to print this prime number, or approximately 22 reams of paper. (15 are shown here.)]]

The record is held by with 41,024,320 digits, found by GIMPS on October 12, 2024.

Prizes

The Electronic Frontier Foundation (EFF) offers several prizes for record primes. In 2008, a ten-million-digit prime won a $100,000 prize and a Cooperative Computing Award from the EFF. Time called this prime the 29th top invention of 2008.

Both these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the $150,000 prize for the first prime of over 100 million digits with the winning participant. A $250,000 prize awaits the first prime with one billion digits.

History

thumb|right|287px|Commemorative postmark used by the [[University of Illinois at Urbana–Champaign|UIUC Math Department after proving that M₁₁₂₁₃ is prime]]

The following table lists the progression of the largest known prime number in ascending order. This is the first time a Mersenne prime has been discovered using GPUs instead of central processing units (CPUs).

{| class="wikitable sortable" border="1"

|+ Largest known prime by year using Cambridge's EDSAC computer

|-

| M₅₂₁

| 157

| 1952

| Raphael M. Robinson

|-

| M₆₀₇

| 183

| 1952

| Raphael M. Robinson

|-

| M₁₂₇₉

| 386

| 1952

| Raphael M. Robinson

|-

| M₂₂₀₃

| 664

| 1952

| Raphael M. Robinson

|-

| M₂₂₈₁

| 687

| 1952

| Raphael M. Robinson

|-

| M₃₂₁₇

| 969

| 1957

| Hans Riesel

|-

| M₄₄₂₃

| 1,332

| 1961

| Alexander Hurwitz

|-

| M₉₆₈₉

| 2,917

| 1963

| Donald B. Gillies

|-

| M₉₉₄₁

| 2,993

| 1963

| Donald B. Gillies

|-

| M₁₁₂₁₃

| 3,376

| 1963

| Donald B. Gillies

|-

| M₁₉₉₃₇

| 6,002

| 1971

| Bryant Tuckerman

|-

| M₂₁₇₀₁

| 6,533

| 1978

| Laura A. Nickel and Landon Curt Noll

|-

| M₂₃₂₀₉

| 6,987

| 1979

| Landon Curt Noll<br />Largest non-Mersenne prime that was the largest known prime when it was discovered.

|-

| M₇₅₆₈₃₉

| 227,832

| 1992

| David Slowinski and Paul Gage of which the largest 20 are listed below.

{| class="wikitable sortable"

! Rank !! Number !! Discovered !! Digits !! Form !!

|-

|style="text-align:right;"| 1

| 2^136279841 − 1

| 2024-10-12

| 41,024,320

| Mersenne

|

|-

|style="text-align:right;"| 3

| 2^77232917 − 1

| 2017-12-26

| 23,249,425

| Mersenne

|

|-

|style="text-align:right;"| 4

| 2^74207281 − 1

| 2016-01-07

| 22,338,618

| Mersenne

|

|-

|style="text-align:right;"| 5

| 2^57885161 − 1

| 2013-01-25

| 17,425,170

| Mersenne

|

|-

|style="text-align:right;"| 6

| 2524190^2097152 + 1

| 2025-10-12

|style="text-align:right;"| 13,426,224

|Generalized Fermat

|

|-

|style="text-align:right;"| 7

| 2^43112609 − 1

| 2008-08-23

| 12,978,189

| Mersenne

|

|-

|style="text-align:right;"| 8

| 2^42643801 − 1

| 2009-06-04

| 12,837,064

| Mersenne

|

|-

|style="text-align:right;"| 9

| Φ₃(−516693^1048576)

| 2023-10-02

| 11,981,518

| Generalized unique

|

|-

|style="text-align:right;"| 10

| Φ₃(−465859^1048576)

| 2023-05-31

| 11,887,192

| Generalized unique

|

|-

|style="text-align:right;"| 11

| 2^37156667 − 1

| 2008-09-06

| 11,185,272

| Mersenne

|

|-

|style="text-align:right;"| 13

| 10223 × 2^31172165 + 1

| 2016-10-31

| 9,383,761

| Proth

|

|-

|style="text-align:right;"| 14

| 2^30402457 − 1

| 2005-12-15

| 9,152,052

| Mersenne

|

|-

|style="text-align:right;"| 15

| 4 × 5^11786358 + 1

| 2024-10-01

| 8,238,312

| Generalized Proth

|

|-

|style="text-align:right;"| 16

| 2^25964951 − 1

| 2005-02-18

| 7,816,230

| Mersenne

|

|-

|style="text-align:right;"| 17

| 4052186 × 69^4052186 + 1

| 2025-04-17

| 7,451,366

| Generalized Cullen

|

|-

|style="text-align:right;"| 18

| 69 × 2^24612729 − 1

| 2024-08-13

| 7,409,102

| Riesel

|

|-

|style="text-align:right;"| 19

| 2^24036583 − 1

| 2004-05-15

| 7,235,733

| Mersenne

|

|-

|style="text-align:right;"| 20

| 5336284^1048576 + 1

| 2025-11-02

| 7,054,022

| Generalized Fermat

|

|}

See also

• List of largest known primes and probable primes

References

External links

• Press release about the former largest known prime 2^74,207,281−1
• Press release about the former largest known prime 2^77,232,917−1
• Press release about the former largest known prime 2^82,589,933−1
• Press release about the largest known prime 2^136,279,841−1