# Mersenne prime
A Mersenne prime is a prime number of the form 2^p − 1, where the exponent p is itself prime; the name honors the French theologian and mathematician [Marin Mersenne](https://www.edgepedia.org/marin-mersenne), who published a list of such exponents in 1644. The largest known prime of any kind is the Mersenne prime 2^136,279,841 − 1, found in October 2024, and the search for these numbers is organized by the [Great Internet Mersenne Prime Search](https://www.edgepedia.org/great-internet-mersenne-prime-search) (GIMPS), a distributed computing project.[1](https://www.britannica.com/science/Mersenne-prime) • [2](https://www.mersenne.org/primes/press/M136279841.html)
| Key fact | Detail |
|---|---|
| Definition | A Mersenne number is 2^p − 1; if it is prime, the exponent p must be prime, though the converse fails (2^11 − 1 = 2047 = 23 × 89)[3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf) |
| Perfect numbers | Every even perfect number has the form 2^(n−1)(2^n − 1) with 2^n − 1 prime, and every Mersenne prime yields one such number[4](https://t5k.org/mersenne/index.html) |
| Largest known | 2^136,279,841 − 1, with 41,024,320 decimal digits, found October 12, 2024 by Luke Durant[2](https://www.mersenne.org/primes/press/M136279841.html) |
| Count known | 52 Mersenne primes are known; over 5.7 million primes lie below 100,000,000, of which only 51 exponents in that range have yielded primes[5](https://cs.uwaterloo.ca/journals/JIS/VOL28/Wagstaff/wag14.pdf) • [3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf) |
| Test | For an odd prime p, the Lucas–Lehmer test decides primality of 2^p − 1 in p − 2 modular squarings, with reduction mod 2^p − 1 done by shift and addition[6](https://simonrs.com/eulercircle/irpw2025/abdur-ll-paper.pdf) |
| Odds per exponent | For a well-trial-factored exponent near 10,000,139, the heuristic prime probability is about 1 in 91,623[7](https://www.mersenne.org/various/math.php) |
| Prizes | The EFF offers $150,000 for the first 100-million-digit prime and $250,000 for the first 1-billion-digit prime, both unclaimed[1](https://www.britannica.com/science/Mersenne-prime) |
## Definition and basic properties
A Mersenne number is 2^p − 1 for an exponent p. If 2^p − 1 is prime, then p must be prime. The proof is a factorization: when n = ab with a, b > 1, the identity (2^a)^b − 1 = (2^a − 1)(2^(a(b−1)) + 2^(a(b−2)) + ... + 2^a + 1) splits 2^n − 1 into two factors that both exceed 1.[3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf) GIMPS exploits this by testing only prime exponents.[7](https://www.mersenne.org/various/math.php)
**The condition is not sufficient.** A prime exponent can still give a composite Mersenne number: 2^11 − 1 = 2047 = 23 × 89.[3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf)
## The Lucas–Lehmer test
Given an odd prime p, the primality of 2^p − 1 can be decided by checking whether 2^p − 1 is a divisor of S(p−1), with S(1) = 4 and S(n+1) = S(n)^2 − 2; put another way, as GIMPS states it, S_(p−2) vanishes for the sequence S_0 = 4, S_N = (S_(N−1)^2 − 2) mod (2^p − 1).[4](https://t5k.org/mersenne/index.html) • [7](https://www.mersenne.org/various/math.php) Édouard Lucas began developing the test in the late 1870s, and [Derrick Henry Lehmer](https://www.edgepedia.org/derrick-henry-lehmer) gave it its rigorous form around 1930.[4](https://t5k.org/mersenne/index.html)
**Why it is fast.** The test performs p − 2 iterations, each dominated by a modular squaring of a p-bit number.[6](https://simonrs.com/eulercircle/irpw2025/abdur-ll-paper.pdf) Reduction modulo 2^p − 1 needs no division: since 2^p ≡ 1 (mod 2^p − 1), a number x = k·2^p + r satisfies x ≡ k + r, computed with a bitwise shift and an addition.[6](https://simonrs.com/eulercircle/irpw2025/abdur-ll-paper.pdf) • [4](https://t5k.org/mersenne/index.html) With schoolbook multiplication the total cost is O(p^3); with Schönhage–Strassen FFT multiplication it drops to roughly O(p^2 log p log log p), which makes testing exponents producing tens of millions of digits feasible.[6](https://simonrs.com/eulercircle/irpw2025/abdur-ll-paper.pdf) GIMPS software uses FFT-based squaring, and the 1994 Crandall–Fagin discrete weighted transform with irrational base more than doubled the squaring speed by allowing a smaller FFT while performing the reduction mod 2^p − 1 for free; the software also checks that convolution roundoff error stays below 0.4.[7](https://www.mersenne.org/various/math.php) A machine-verified, executable version of the test has been formalized in the Isabelle Archive of Formal Proofs.[8](https://isa-afp.org/browser_info/current/AFP/Mersenne_Primes/document.pdf)
## Connection to perfect numbers
According to the Euclid–Euler theorem, an even number k is perfect exactly when it can be written as 2^(n−1)(2^n − 1), where 2^n − 1 is prime.[4](https://t5k.org/mersenne/index.html) The correspondence is one-to-one for even perfect numbers: every Mersenne prime 2^p − 1 yields the even perfect number 2^(p−1) × (2^p − 1), and every even perfect number arises this way.[9](https://theconversation.com/a-41-million-digit-prime-number-is-the-biggest-ever-found-but-mathematicians-search-for-perfection-will-continue-242291) The newly discovered prime yields the perfect number 2^136,279,840 × (2^136,279,841 − 1), which is over 82 million digits long.[2](https://www.mersenne.org/primes/press/M136279841.html) The correspondence covers only even perfect numbers; it is not known whether any odd perfect numbers exist.[2](https://www.mersenne.org/primes/press/M136279841.html) • [9](https://theconversation.com/a-41-million-digit-prime-number-is-the-biggest-ever-found-but-mathematicians-search-for-perfection-will-continue-242291)
## Known Mersenne primes and the GIMPS search
Fifty-two [Mersenne primes](https://www.edgepedia.org/mersenne-primes) are known.[5](https://cs.uwaterloo.ca/journals/JIS/VOL28/Wagstaff/wag14.pdf) George Woltman began consolidating the scattered search databases and placing a free optimized program on the web in late 1995, and GIMPS was founded in 1996 as one of the pioneers in creating supercomputer-scale power from inexpensive personal computers; volunteers have discovered every Mersenne prime found since late 1996.[4](https://t5k.org/mersenne/index.html) • [10](https://mathstat.dal.ca/FQ/Papers1/46_47-3/Woltman_Kurowski.pdf) • [11](https://mathworld.wolfram.com/MersennePrime.html) Britannica counts more than 150,000 volunteers running the project's software.[1](https://www.britannica.com/science/Mersenne-prime)
**The testing pipeline.** Before primality testing, GIMPS runs trial factoring and P-1 factoring to eliminate candidates; the chance of finding a factor between 2^x and 2^(x+1) is about 1/x.[7](https://www.mersenne.org/various/math.php) Since 2018 the first-time primality check has been a Fermat probable prime (PRP) test with Robert Gerbicz's error checking, which reduced the chance of missing a new Mersenne prime from about 1–2% to extremely unlikely; in 2020, PRP proofs based on Krzysztof Pietrzak's method, verifiable over 100 times faster than a standard double-check, became the preferred search method.[7](https://www.mersenne.org/various/math.php) A successful candidate is then confirmed with several independent Lucas–Lehmer tests on different hardware and software.[2](https://www.mersenne.org/primes/press/M136279841.html) Historically, a Lucas–Lehmer test with no reported errors had an error rate around 1.5%, versus roughly 50% when an error was reported, and double-checks typically occur about 8 years after first-time tests.[7](https://www.mersenne.org/various/math.php)
**How rare a hit is.** For an exponent like 10,000,139 with trial factoring complete to 2^64, the rigorous heuristic prime probability is (how_far_factored − 1) / (exponent × Euler's constant), about 1 in 91,623.[7](https://www.mersenne.org/various/math.php) The scarcity is visible in the totals: over 5.7 million primes lie below 100,000,000, and only 51 exponents in that range have produced prime Mersenne values.[3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf)
## History
In 1644, in the preface of *Cogitata Physica-Mathematica*, Mersenne asserted that 2^n − 1 is prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 and composite for the other 44 primes below 257.[1](https://www.britannica.com/science/Mersenne-prime) • [12](https://mathshistory.st-andrews.ac.uk/SH/mersenne_sh.pdf) The list contained five errors: 67 and 257 are composite, while 61, 89, and 107 belong in it.[11](https://mathworld.wolfram.com/MersennePrime.html) The first error identified was that 61 should be in the list, a fact not discovered before 1883, and the corrected list (2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127) was not fully determined until 1947.[13](https://www.tau.ac.il/~corry/publications/articles/pdf/Mersenne%20-%20Computers%20-%20Rutherford.pdf) • [4](https://t5k.org/mersenne/index.html)
**Correcting the list in public.** In 1903 Frank Nelson Cole presented the factorization of 2^67 − 1 to the American Mathematical Society by silently working the two sides of the calculation on a blackboard, then sat down to applause.[11](https://mathworld.wolfram.com/MersennePrime.html) In 1952 Raphael Robinson used the SWAC computer to find five Mersenne primes with exponents 521, 607, 1279, 2203, and 2281; the largest had 687 decimal digits, an early case of an electronic computer extending the largest-known-prime record.[11](https://mathworld.wolfram.com/MersennePrime.html)
## The 2024 discovery: recent developments
The 52nd known Mersenne prime, 2^136,279,841 − 1, was found by Luke Durant on October 12, 2024.[2](https://www.mersenne.org/primes/press/M136279841.html) On October 11 an NVIDIA A100 GPU in Dublin, Ireland reported the number probably prime; on October 12 an NVIDIA H100 in [San Antonio](https://www.edgepedia.org/san-antonio), Texas confirmed primality with a Lucas–Lehmer test.[2](https://www.mersenne.org/primes/press/M136279841.html) Durant, a former Nvidia programmer, assembled a cloud supercomputer of thousands of server GPUs spanning 24 datacenter regions in 17 countries.[2](https://www.mersenne.org/primes/press/M136279841.html) • [14](https://www.washingtonpost.com/science/2024/10/23/nvidia-prime-mersenne-gpu-cloud/) The key software step came in 2017, when Mihai Preda wrote the GpuOwl program to test Mersenne numbers on GPUs.[2](https://www.mersenne.org/primes/press/M136279841.html)
Independent verification used multiple programs: Prime95 on Intel CPUs, PRPLL/GpuOwl on AMD and NVIDIA GPUs, Mlucas (confirming on October 19), and CUDALucas.[2](https://www.mersenne.org/primes/press/M136279841.html) The discovery ended the 28-year era in which ordinary personal computers found these primes, and it was the first Mersenne prime found since 2018.[2](https://www.mersenne.org/primes/press/M136279841.html) • [15](https://www.npr.org/2024/11/07/nx-s1-5170260/amateur-mathematician-finds-the-new-largest-prime-number) The shift to PRP testing with verifiable proofs on GPU hardware is what made a cloud-scale search practical.[7](https://www.mersenne.org/various/math.php)
## By the numbers
The largest known prime has 41,024,320 decimal digits.[2](https://www.mersenne.org/primes/press/M136279841.html) Its associated perfect number exceeds 82 million digits.[2](https://www.mersenne.org/primes/press/M136279841.html) More than 5.7 million prime exponents lie below 100 million, and 51 of them have yielded Mersenne primes; 52 Mersenne primes are known in total.[3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf) • [5](https://cs.uwaterloo.ca/journals/JIS/VOL28/Wagstaff/wag14.pdf) A single well-trial-factored exponent near 10 million carries about a 1-in-91,623 chance of being prime.[7](https://www.mersenne.org/various/math.php) The Electronic Frontier Foundation established prizes for the first verified prime with more than 1 million digits ($50,000, awarded 2006), 10 million digits ($100,000, awarded 2008), 100 million digits ($150,000), and 1 billion digits ($250,000); the last two remain unclaimed, and the $150,000 award is funded by an anonymous donor. GIMPS also offers a $3,000 research discovery award per Mersenne prime.[1](https://www.britannica.com/science/Mersenne-prime) • [2](https://www.mersenne.org/primes/press/M136279841.html)
## Distribution conjectures
It is not known whether the set of Mersenne primes is finite or infinite, though infinitely many are conjectured.[16](https://oeis.org/wiki/Mersenne_prime) • [3](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf) The Lenstra–Pomerance–Wagstaff conjecture asserts there are infinitely many Mersenne primes and predicts their order of growth.[16](https://oeis.org/wiki/Mersenne_prime) A 1989 conjecture, the New Mersenne Conjecture, states that for an odd prime p, if any two of three statements hold (p has the form 2^k ± 1 or 4k ± 3; M_p is prime; (2^p + 1)/3 is a Wagstaff prime), then the third also holds; Samuel Wagstaff's 2025 paper examines a new conjecture made that year and gives evidence that both are probably true.[5](https://cs.uwaterloo.ca/journals/JIS/VOL28/Wagstaff/wag14.pdf)
## References
1. [Mersenne prime, Encyclopaedia Britannica](https://www.britannica.com/science/Mersenne-prime)
2. [52nd Known Mersenne Prime Discovered, GIMPS press release](https://www.mersenne.org/primes/press/M136279841.html)
3. [The Lucas–Lehmer Test, Keith Conrad, University of Connecticut](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/lucas-lehmer-test.pdf)
4. [Mersenne Primes: History, Theorems and Lists, PrimePages](https://t5k.org/mersenne/index.html)
5. [Two Mersenne Prime Conjectures, Journal of Integer Sequences (Wagstaff)](https://cs.uwaterloo.ca/journals/JIS/VOL28/Wagstaff/wag14.pdf)
6. [A Comprehensive Exposition of the Lucas-Lehmer Primality Test](https://simonrs.com/eulercircle/irpw2025/abdur-ll-paper.pdf)
7. [GIMPS – The Math – PrimeNet](https://www.mersenne.org/various/math.php)
8. [Mersenne primes and the Lucas–Lehmer test, Isabelle Archive of Formal Proofs](https://isa-afp.org/browser_info/current/AFP/Mersenne_Primes/document.pdf)
9. [A 41-million-digit prime number is the biggest ever found, The Conversation](https://theconversation.com/a-41-million-digit-prime-number-is-the-biggest-ever-found-but-mathematicians-search-for-perfection-will-continue-242291)
10. [GIMPS paper, Fibonacci Quarterly (Woltman & Kurowski)](https://mathstat.dal.ca/FQ/Papers1/46_47-3/Woltman_Kurowski.pdf)
11. [Mersenne Prime, Wolfram MathWorld](https://mathworld.wolfram.com/MersennePrime.html)
12. [Marin Mersenne – Mathematics and Universal Harmony, MacTutor](https://mathshistory.st-andrews.ac.uk/SH/mersenne_sh.pdf)
13. [Hunting Prime Numbers from Human to Electronic Computers](https://www.tau.ac.il/~corry/publications/articles/pdf/Mersenne%20-%20Computers%20-%20Rutherford.pdf)
14. [New largest prime number found by ex-Nvidia employee, The Washington Post](https://www.washingtonpost.com/science/2024/10/23/nvidia-prime-mersenne-gpu-cloud/)
15. [Amateur mathematician finds the new largest prime number, NPR](https://www.npr.org/2024/11/07/nx-s1-5170260/amateur-mathematician-finds-the-new-largest-prime-number)
16. [Mersenne prime, OeisWiki](https://oeis.org/wiki/Mersenne_prime)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Elementary number theory › Prime numbers: elementary aspects*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 11, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license