# Definition:Mersenne Prime

## Contents

## Definition

A **Mersenne prime** is a Mersenne number which happens to be prime.

That is, it is a prime number of the form $2^p - 1$.

The number $2^p - 1$ is, in this context, often denoted $M_p$.

### Index of Mersenne Prime

The **index** of the **Mersenne prime** $M_p = 2^p - 1$ is the (prime) number $p$.

## History of Mersenne Primes

Mersenne primes are named for Marin Mersenne, who published a book *Cogitata Physico-Mathematica* in $1644$, in which he claimed that the only primes $p \le 257$ for which $2^p - 1$ is prime are $2, 3, 5, 7, 13, 17, 19, 31, 67, 127$ and $257$.

He was not entirely correct, as shall be seen.

Previous to that, the special nature of these primes had been noted by Euclid, who showed that if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect. The first four primes of this form were known to him.

The fifth one, $M_{13}$, may have been known to Iamblichus in the $4$th century A.D.^{[1]}, but this is uncertain, as he does not explicitly demonstrate it. It was definitely known about by $1456$.

Hudalrichus Regius showed in $1536$ that not all numbers of the form $2^n - 1$ for odd $n$ are prime numbers.

Up until this time it was assumed so, but Regius gave the counterexamples $2^9 - 1 = 511 = 7 \times 73$ and $2^{11} - 1 = 2047 = 23 \times 89$.

Hence he also demonstrated that it is not even sufficient for $n$ to be prime for $2^n - 1$ also to be prime.

Pietro Cataldi is supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$. Recent researches^{[2]}, however, suggest that these may have already been discovered by $1460$. But as no evidence has been found from that date that they had been proven to be prime, it is possible that these were just lucky guesses.

Cataldi also claimed the primality of the Mersenne numbers $M_{23}, M_{29}, M_{31}$ and $M_{37}$.^{[3]} In this he was correct only about $M_{31}$, so that also backs up the suggestion that he was only guessing.

Work started in earnest on these numbers from Mersenne's work.

- $1640$: Fermat showed that $M_{23}$ has $47$ as a divisor, and that $M_{37}$ has $223$ as a divisor.

- $1811$: Peter Barlow (somewhat short-sightedly, given historical 20-20 hindsight) stated in his book
*Elementary Investigation of the Theory of Numbers*that $M_{31}$:

*is the greatest [i.e. prime number] at present known to be such ... and probably the greatest that ever will be discovered.*See Barlow's Prediction.

- $1876$: François Édouard Anatole Lucas proved that $M_{127}$ is prime, and also discovered that $M_{67}$ is actually composite.

- $1883$: Ivan Pervushin proved that $M_{61}$ is prime.

- $1903$: The factors of $M_{67}$ were found by Frank Cole who delivered a now famous lecture
*On The Factorization of Large Numbers*in which he performed (without uttering a word) the arithmetic demonstrating what those factors were.

- $1911$: R.E. Powers proved that $M_{89}$ is prime.

- $1914$: R.E. Powers proved that $M_{107}$ is prime, although precedence for this is also claimed by E. Fauquembergue.

- $1916$: R.E. Powers proved that $M_{241}$ is composite.

- $1922$: Maurice Kraitchik proved that $M_{257}$ is actually composite.

Thus Mersenne's assertion was finally investigated in full: he had been determined to be wrong by:

- including $M_{67}$ and $M_{257}$ in his list of primes;
- failing to include $M_{61}$, $M_{89}$ and $M_{107}$.

(Pervushin's discovery of the primality of $M_{61}$ caused some to suggest that Mersenne's claim of the primality of $M_{67}$ may have been a copying error for $M_{61}$.)

Nobody will ever know how Mersenne came to his conclusions, as it is impossible with the mathematical knowledge of the time for him to have worked it all out by hand. The fact that he made so few mistakes is incredible.

The work continued, and does so to this day.^{[4]}

- $1952$: Raphael Robinson used a computer to show that $M_{521}, M_{607}, M_{1279}, M_{2203}$ and $M_{2281}$ are all prime.

- During the next four decades, the count of known Mersenne primes was doubled by various mathematicians testing supercomputers.

Since then, hunting for Mersenne primes has become a casual hobby for anyone who has access to a computer.

## Testing Primality of a Mersenne number

The Lucas-Lehmer Test is a way of determining the primality of a given $M_p$ without laboriously testing each possible prime divisor.

## G.I.M.P.S.

The **Great Internet Mersenne Prime Search**, known as **G.I.M.P.S.** (or just **GIMPS**), has become a gathering place for Number Theorists interested in the discovery of the Mersenne primes.

It was founded by George F. Woltman in early $1996$.

All the Mersenne primes discovered from November $1996$ onwards have been discovered by this team.

In particular August and September of $2008$ alone, both the $45$th and $46$th Mersenne primes were discovered:

$M_{43,112,609}$ (a $12,978,189$ digit number)

and

$M_{37,156,667}$ (an $11,185,272$ digit number)

becoming the first Mersenne primes of $10$ million digits to be found.

You can follow the work of G.I.M.P.S. at www.mersenne.org.

## Currently known Mersenne Primes

Prime $p$ | Prime $M_p$ | Number of digits in $M_p$ | Date discovered | Discovered by | |
---|---|---|---|---|---|

1 | $ 2 $ | $ 3 $ | $ 1 $ | Known to Euclid | |

2 | $ 3 $ | $ 7 $ | $ 1 $ | Known to Euclid | |

3 | $ 5 $ | $ 31 $ | $ 2 $ | Known to Euclid | |

4 | $ 7 $ | $ 127 $ | $ 3 $ | Known to Euclid | |

5 | $ 13 $ | $ 8191 $ | $ 4 $ | 1456 | |

6 | $ 17 $ | $ 131 \, 071 $ | $ 6 $ | 1588 | Pietro Antonio Cataldi |

7 | $ 19 $ | $ 524 \, 287 $ | $ 6 $ | 1588 | Pietro Antonio Cataldi |

8 | $ 31 $ | $ 2 \, 147 \, 483 \, 647 $ | $ 10 $ | 1772 | Leonhard Paul Euler |

9 | $ 61 $ | $ 2 \cdotp 305 \times 10^{18} $ | $ 19 $ | 1883 | Ivan Mikheevich Pervushin |

10 | $ 89 $ | $ 6 \cdotp 189 \times 10^{26} $ | $ 27 $ | 1911 | R.E. Powers |

11 | $ 107 $ | $ 1 \cdotp 622 \times 10^{32} $ | $ 33 $ | 1914 | R.E. Powers |

12 | $ 127 $ | $ 1 \cdotp 701 \times 10^{38} $ | $ 39 $ | 1876 | François Édouard Anatole Lucas |

13 | $ 521 $ | $ 6 \cdotp 865 \times 10^{156} $ | $ 157 $ | 30 Jan 1952 | Raphael Mitchel Robinson |

14 | $ 607 $ | $ 5 \cdotp 311 \times 10^{182} $ | $ 183 $ | 30 Jan 1952 | Raphael Mitchel Robinson |

15 | $ 1279 $ | $ 1 \cdotp 041 \times 10^{385} $ | $ 386 $ | 25 Jun 1952 | Raphael Mitchel Robinson |

16 | $ 2203 $ | $ 1 \cdotp 476 \times 10^{663} $ | $ 664 $ | 7 Oct 1952 | Raphael Mitchel Robinson |

17 | $ 2281 $ | $ 4 \cdotp 461 \times 10^{686} $ | $ 687 $ | 9 Oct 1952 | Raphael Mitchel Robinson |

18 | $ 3217 $ | $ 2 \cdotp 591 \times 10^{968} $ | $ 969 $ | 8 Sept 1957 | Hans Ivar Riesel |

19 | $ 4253 $ | $ 1 \cdotp 908 \times 10^{1280} $ | $ 1281 $ | 3 Nov 1961 | Alexander Hurwitz |

20 | $ 4423 $ | $ 2 \cdotp 855 \times 10^{1331} $ | $ 1332 $ | 3 Nov 1961 | Alexander Hurwitz |

21 | $ 9689 $ | $ 4 \cdotp 782 \times 10^{2916} $ | $ 2917 $ | 11 May 1963 | Donald Bruce Gillies |

22 | $ 9941 $ | $ 3 \cdotp 461 \times 10^{2992} $ | $ 2993 $ | 16 May 1963 | Donald Bruce Gillies |

23 | $ 11 \, 213 $ | $ 2 \cdotp 814 \times 10^{3375} $ | $ 3376 $ | 2 Jun 1963 | Donald Bruce Gillies |

24 | $ 19 \, 937 $ | $ 4 \cdotp 315 \times 10^{6001} $ | $ 6002 $ | 4 Mar 1971 | Bryant Tuckerman |

25 | $ 21 \, 701 $ | $ 4 \cdotp 487 \times 10^{6532} $ | $ 6533 $ | 30 Oct 1978 | Landon Curt Noll and Ariel Nickel |

26 | $ 23 \, 209 $ | $ 4 \cdotp 029 \times 10^{6986} $ | $ 6987 $ | 9 Feb 1979 | Landon Curt Noll |

27 | $ 44 \, 497 $ | $ 8 \cdotp 545 \times 10^{13 \, 394} $ | $ 13 \, 395 $ | 8 Apr 1979 | Harry Lewis Nelson and David Slowinski |

28 | $ 86 \, 243 $ | $ 5 \cdotp 369 \times 10^{25 \, 961} $ | $ 25 \, 962 $ | 25 Sept 1982 | David Slowinski |

29 | $ 110 \, 503 $ | $ 5 \cdotp 219 \times 10^{33 \, 264} $ | $ 33 \, 265 $ | 28 Jan 1988 | Walt Colquitt and Luke Welsh |

30 | $ 132 \, 049 $ | $ 5 \cdotp 127 \times 10^{39 \, 750} $ | $ 39 \, 751 $ | 19 Sept 1983 | David Slowinski |

31 | $ 216 \, 091 $ | $ 7 \cdotp 461 \times 10^{65 \, 049} $ | $ 65 \, 050 $ | 1 Sept 1985 | David Slowinski |

32 | $ 756 \, 839 $ | $ 1 \cdotp 741 \times 10^{227 \, 831} $ | $ 227 \, 832 $ | 19 Feb 1992 | David Slowinski and Paul Gage |

33 | $ 859 \, 433 $ | $ 1 \cdotp 295 \times 10^{258 \, 715} $ | $ 258 \, 716 $ | 4 Jan 1994 | David Slowinski and Paul Gage |

34 | $ 1 \, 257 \, 787 $ | $ 4 \cdotp 122 \times 10^{378 \, 631} $ | $ 378 \, 632 $ | 3 Sept 1996 | David Slowinski and Paul Gage |

35 | $ 1 \, 398 \, 269 $ | $ 8 \cdotp 147 \times 10^{420 \, 920} $ | $ 420 \, 921 $ | 13 Nov 1996 | GIMPS / Joel Armengaud |

36 | $ 2 \, 976 \, 221 $ | $ 6 \cdotp 233 \times 10^{895 \, 931} $ | $ 895 \, 932 $ | 24 Aug 1997 | GIMPS / Gordon Spence |

37 | $ 3 \, 021 \, 377 $ | $ 1 \cdotp 274 \times 10^{909 \, 525} $ | $ 909 \, 526 $ | 27 Jan 1998 | GIMPS / Roland Clarkson |

38 | $ 6 \, 972 \, 593 $ | $ 4 \cdotp 371 \times 10^{2 \, 098 \, 959} $ | $ 2 \, 098 \, 960 $ | 1 Jun 1999 | GIMPS / Nayan Hajratwala |

39 | $ 13 \, 466 \, 917 $ | $ 9 \cdotp 249 \times 10^{4 \, 053 \, 945} $ | $ 4 \, 053 \, 946 $ | 14 Nov 2001 | GIMPS / Michael Cameron |

40 | $ 20 \, 996 \, 011 $ | $ 1 \cdotp 260 \times 10^{6 \, 320 \, 429} $ | $ 6 \, 320 \, 430 $ | 17 Nov 2003 | GIMPS / Michael Shafer |

41 | $ 24 \, 036 \, 583 $ | $ 2 \cdotp 994 \times 10^{7 \, 235 \, 732} $ | $ 7 \, 235 \, 733 $ | 15 May 2004 | GIMPS / Josh Findley |

42 | $ 25 \, 964 \, 951 $ | $ 1 \cdotp 222 \times 10^{7 \, 816 \, 229} $ | $ 7 \, 816 \, 230 $ | 18 Feb 2005 | GIMPS / Martin Nowak |

43 | $ 30 \, 402 \, 457 $ | $ 3 \cdotp 154 \times 10^{9 \, 152 \, 051} $ | $ 9 \, 152 \, 052 $ | 15 Dec 2005 | GIMPS / Curtis Cooper and Steven Boone |

44 | $ 32 \, 582 \, 657 $ | $ 1 \cdotp 246 \times 10^{9 \, 808 \, 358} $ | $ 9 \, 808 \, 358 $ | 4 Sept 2006 | GIMPS / Curtis Cooper and Steven Boone |

45 | $ 37 \, 156 \, 667 $ | $ 2 \cdotp 023 \times 10^{11 \, 185 \, 271} $ | $ 11 \, 185 \, 272 $ | 6 Sept 2008 | GIMPS / Hans-Michael Elvenich |

46 | $ 42 \, 643 \, 801 $ | $ 1 \cdotp 699 \times 10^{12 \, 837 \, 063} $ | $ 12 \, 837 \, 064 $ | 12 Apr 2009 | GIMPS / Odd Magnar Strindmo |

47 | $ 43 \, 112 \, 609 $ | $ 3 \cdotp 165 \times 10^{12 \, 978 \, 188} $ | $ 12 \, 978 \, 189 $ | 23 Aug 2008 | GIMPS / Edson Smith |

$ 57 \, 885 \, 161 $ | $ 5 \cdotp 818 \times 10^{17 \, 425 \, 169} $ | $ 17 \, 425 \, 170 $ | 25 Jan 2013 | GIMPS / Curtis Cooper | |

$ 74 \, 207 \, 281 $ | $ 3 \cdotp 003 \times 10^{22 \, 338 \, 617} $ | $ 22 \, 338 \, 618 $ | 07 Jan 2016 | GIMPS / Curtis Cooper | |

$ 77 \, 232 \, 917 $ | $ 4 \cdotp 673 \times 10^{23 \, 249 \, 424} $ | $ 23 \, 249 \, 425 $ | 26 Dec 2017 | GIMPS / Jon Pace | |

$ 82 \, 589 \, 933 $ | $ 1 \cdotp 488 \times 10^{24 \, 862 \, 047} $ | $ 24 \, 862 \, 048 $ | 07 Dec 2018 | GIMPS / Patrick Laroche |

Note that the index numbers of Mersenne primes after no. $47$ are uncertain, as there may still be undiscovered Mersenne primes between the $47$th and $51$st.

Not all numbers in that range have been explored yet.

## Also see

- Definition:Mersenne Number
- Theorem of Even Perfect Numbers
- Primes of form Power Less One: for $2^p - 1$ to be prime, then $p$ must also be prime.
- Do Mersenne Primes Go On For Ever?

- Results about
**Mersenne primes**can be found here.

## Source of Name

This entry was named for Marin Mersenne.

## References

- ↑ 1908: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements*: footnote to Book $\text{IX}$ Proposition $36$. - ↑ Ettore Picutti, in
*Historia Mathematica*pp $123$ - $136$ ($1989$), records that an anonymous author had discovered by $1460$ that both $M_{17}$ and $M_{19}$ are prime. - ↑ See The Largest Known Prime by Year.
- ↑ See Chris Caldwell's website The Prime Pages for more detail of the history of the unearthing of Mersenne primes, and more.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $7$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $28$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $127$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.12$: Mersenne ($\text {1588}$ – $\text {1648}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $28$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $127$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Mersenne numbers** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Mersenne numbers** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Euclid - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Mersenne prime**