A table ordered by discovery date follows:

Year | exponent | discoverer (credited persons) | statement | method | comment |
---|---|---|---|---|---|

Antique | p | - | conjecture | guess | all mersenne numbers with prime exponent are primes |

Antique | 2 | - | proved prime | TD | confirm conjecture |

Antique | 3 | - | proved prime | TD | confirm conjecture |

Antique | 5 | - | proved prime | TD | confirm conjecture |

Antique | 7 | - | proved prime | TD | confirm conjecture |

1456 | 13 | Codex Lat. Monac 14908 | proved prime | TD | 23 trial divisions |

1536 | 11 | Hudalricus Regius | factor 23 published | TD | disproof that all mersenne numbers are prime |

1588 | 17 | Pietro Cataldi | proved prime | TD | 71 trial divisions |

1588 | 19 | Pietro Cataldi | proved prime | TD | 127 trial divisions |

1640 | 23 | Pierre de Fermat | smallest factor 47 found | FD | conjectured factor form: d = 2pk+1 |

1640 | 37 | Pierre de Fermat | smallest factor 223 found | FD | conjectured factor form: d = 2pk+1 |

1644 | p | Marin Mersenne | conjecture | guess | conjectured all Mersenne primes <= 257 have one of these exponents: 2,3,5,7,13,17,19,31,67,127,257 |

1732 | 29 | Leonhard Euler | smallest factor 233 found | FD | proved factor form; 2 trial divisions |

1732 | 43 | Leonhard Euler | smallest factor 431 found | FD | proved factor form; 2 trial divisions |

1732 | 73 | Leonhard Euler | smallest factor 439 found | FD | proved factor form; 2 trial divisions |

1732 | 83 | Leonhard Euler | smallest factor 167 found | ES | proved special factor theorem |

1732 | 131 | Leonhard Euler | smallest factor 263 found | ES | proved special factor theorem |

1732 | 179 | Leonhard Euler | smallest factor 359 found | ES | proved special factor theorem |

1732 | 191 | Leonhard Euler | smallest factor 383 found | ES | proved special factor theorem |

1732 | 239 | Leonhard Euler | smallest factor 479 found | ES | proved special factor theorem |

1732 | 251 | Leonhard Euler | smallest factor 503 found | ES | proved special factor theorem |

1741 | 47 | Leonhard Euler | smallest factor 2351 found | FE | proved preciser factor form: 7 trial divisions |

1750 | 31 | Leonhard Euler | proved prime | FE | proved preciser factor form: 84 trial divisions |

1856 | 79 | Carl Gustav Reuschle | smallest factor published | FE | 2 trial divisions |

1856 | 113 | Carl Gustav Reuschle | smallest factor published | FE | 3 trial divisions |

1856 | 233 | Carl Gustav Reuschle | smallest factor published | FE | 2 trial divisions |

1859 | 41 | Giovanni Antonio Amedeo Plana | smallest factor found | FE | 17 trial divisions |

1867 | 53 | Fortune Landry | smallest factor found | FE | 6 trial divisions |

1876 | p | Francois Edouard A. Lucas | Lucas Test formulated | LT | efficient iterative test |

1876 | 67 | Francois Edouard A. Lucas | disproved(?) prime | LT | 65 iterations -- calculation was never checked |

1876 | 127 | Francois Edouard A. Lucas | proved prime | LT | 125 iterations |

1878 | 59 | Fortune Landry | smallest factor found | FE | 143 trial divisions |

1883 | 61 | Ivane M. Pervouchine | proved prime | LT | 59 iterations |

1883 | 97 | H. Le Lasseur | smallest factor found | FE | 7 trial divisions |

1883 | 151 | H. Le Lasseur | smallest factor found | FE | 8 trial divisions |

1883 | 211 | H. Le Lasseur | smallest factor found | FE | 4 trial divisions |

1883 | 223 | H. Le Lasseur | smallest factor found | FE | 3 trial divisions |

1895 | 197 | Allan J. C. Cunningham | smallest factor found | FE | 2 trial divisions |

1908 | 163 | Allan J. C. Cunningham | smallest factor found | FE | 42 trial divisions |

1909 | 71 | Allan J. C. Cunningham | smallest factor found | FE | 150 trial divisions |

1911 | 89 | R. E. Powers | proved prime | LT | 87 iterations |

1911 | 181 | Herbert J. Woodall | smallest factor found | FE | 15 trial divisions |

1912 | 173 | Allan J. C. Cunningham | smallest factor found | FE | ? trial divisions |

1914 | 103 | R. E. Powers | disproved(?) prime | LT | 101 iterations -- calculation was never checked |

1914 | 107 | R. E. Powers | proved prime | LT | 105 iterations |

1914 | 109 | R. E. Powers | disproved(?) prime | LT | 107 iterations -- calculation was never checked |

1926 | 139 | Derrick H. Lehmer | disproved prime | LL | 137 iterations |

1932 | 149 | Derrick H. Lehmer | disproved prime | LL | 147 iterations |

1932 | 257 | Derrick H. Lehmer | disproved prime | LL | 255 iterations |

1934 | 241 | R. E. Powers | disproved prime | LL | 239 iterations |

1944 | 157 | Horace. S. Uhler | disproved prime | LL | 155 iterations |

1944 | 167 | Horace S. Uhler | disproved prime | LL | 165 iterations |

1944 | 193 | Horace S. Uhler | disproved prime | LL | 191 iterations |

1945 | 229 | Horace S. Uhler | disproved prime | LL | 227 iterations |

1946-07-27 | 199 | Horace S. Uhler | disproved prime | LL | 197 iterations |

1947-06-04 | 227 | Horace S. Uhler | disproved prime | LL | 225 iterations |

1947 | p <= 257 | - | Mersenne's conjecture believed to be checked | FE/LL | 101 & 137 were wronlgy checked or claimed by E. Fauquembergue in 1914 and 1920 to be prime |

1952 | p <= 257 | Raphael M. Robinson | checked Mersenne numbers | LL | successfully used first computer for Mersenne search |

1952 | 101 | Raphael M. Robinson | disproved prime | LL | showed E. Fauquembergue miscalculated |

1952 | 137 | Raphael M. Robinson | disproved prime | LL | showed E. Fauquembergue miscalculated |

1952-01-30 | 521 | Raphael M. Robinson | proved prime | LL | SWAC used |

1952-01-30 | 607 | Raphael M. Robinson | proved prime | LL | SWAC used |

1952-06-25 | 1279 | Raphael M. Robinson | proved prime | LL | SWAC used |

1953 | p < 2304 | Raphael M. Robinson | checked Meresenne numbers | LL | SWAC used |

1957 | 8191 | David J. Wheeler | disproved prime | LL | Illiac I used |

1957 | 2300 < p < 3300 | Hans Riesel | checked Meresenne numbers | LL | BESK used |

1957-09-08 | 3217 | Hans Riesel | proved prime | LL | BESK used |

1961 | 3300 < p < 5000 | Alexander Hurwitz | checked Meresenne numbers | LL | IBM 7090 used |

1961-11-03 | 4253 | Alexander Hurwitz | proved prime | LL | IBM 7090 used |

1961-11-03 | 4423 | Alexander Hurwitz | proved prime | LL | IBM 7090 used |

1962 | 5000 < p < 6000 | Alexander Hurwitz | checked Meresenne numbers | LL | IBM 7090 used |

1963 | 6000 < p < 7000 | Sidney Kravitz and Murray Berg | checked Meresenne numbers | LL | IBM 7090 used |

1983 | p | Samuel S. Wagstaff, H. W. Lenstra, Carl B. Pomerance | conjecture | guess | the number of Mersenne primes up to the limit N is about C ln ln N, with C= e^{g}/ ln 2 |

- TD trial division
- to prove a number n to be a prime number, check whether n is not divisable by all primes p with p
^{2}<= n. - FD Fermat's trial division
- to find a factor of 2
^{p}-1 try only factors of the form 2kp+1 for positive integers k - ES Euler's special divisor theorem
- If p= 4m-1 and 2p+1=8m-1 are primes, then 2
^{p}-1 is divisible by 2p+1 - FE Euler's improved divisor theorem
- If d divides 2
^{p}-1, then d = 2kp+1 for a positive integer and d = +/- 1 (mod 8) - LT Lucas 's Test
- If p=4k+3, then let S
_{0}=4 else S_{0}=3. And for i>0: S_{i}= S_{i-1}^{2}-2 . If and only if S_{p-2}= 0 (mod 2^{p}-1) then 2^{p}-1 is a prime number. - LL Lucas-Lehmer Test
- Let S
_{0}=4 and for i>0: S_{i}= S_{i-1}^{2}-2 (mod 2^{p}-1). If and only if S_{p-2}= 0 , then 2^{p}-1 is a prime number.

- Publication of Lucas-Test by E. Lucas 1891
- More details on History of Mersenne Primes by R.C. Archibald 1935
- First results of computer checking with the LL-Test by R.M. Robinson 1954
- GIMPS (Distributed search for Mersenne Primes

Achim Flammenkamp

2006-05-04 23:12 UTC+2