For which prime numbers p is 2p-1 a prime number?

Such prime numbers are called now-a-days Mersenne Primes and p is called its exponent.
A table ordered by discovery date follows:

Year exponent discoverer (credited persons) statement method comment
Antiquep-conjectureguessall mersenne numbers with prime exponent are primes
Antique2-proved primeTDconfirm conjecture
Antique3-proved primeTDconfirm conjecture
Antique5-proved primeTDconfirm conjecture
Antique7-proved primeTDconfirm conjecture
145613Codex Lat. Monac 14908proved primeTD 23 trial divisions
153611Hudalricus Regiusfactor 23 publishedTDdisproof that all mersenne numbers are prime
158817Pietro Cataldiproved primeTD 71 trial divisions
158819Pietro Cataldiproved primeTD 127 trial divisions
164023Pierre de Fermatsmallest factor 47 foundFDconjectured factor form: d = 2pk+1
164037Pierre de Fermatsmallest factor 223 foundFDconjectured factor form: d = 2pk+1
1644pMarin Mersenneconjectureguessconjectured all Mersenne primes <= 257 have one of these exponents: 2,3,5,7,13,17,19,31,67,127,257
173229Leonhard Eulersmallest factor 233 foundFDproved factor form; 2 trial divisions
173243Leonhard Eulersmallest factor 431 foundFDproved factor form; 2 trial divisions
173273Leonhard Eulersmallest factor 439 foundFDproved factor form; 2 trial divisions
173283Leonhard Eulersmallest factor 167 foundESproved special factor theorem
1732131Leonhard Eulersmallest factor 263 foundESproved special factor theorem
1732179Leonhard Eulersmallest factor 359 foundESproved special factor theorem
1732191Leonhard Eulersmallest factor 383 foundESproved special factor theorem
1732239Leonhard Eulersmallest factor 479 foundESproved special factor theorem
1732251Leonhard Eulersmallest factor 503 foundESproved special factor theorem
174147Leonhard Eulersmallest factor 2351 foundFEproved preciser factor form: 7 trial divisions
175031Leonhard Eulerproved primeFEproved preciser factor form: 84 trial divisions
185679Carl Gustav Reuschlesmallest factor publishedFE2 trial divisions
1856113Carl Gustav Reuschlesmallest factor publishedFE3 trial divisions
1856233Carl Gustav Reuschlesmallest factor publishedFE2 trial divisions
185941Giovanni Antonio Amedeo Planasmallest factor foundFE17 trial divisions
186753Fortune Landrysmallest factor foundFE6 trial divisions
1876pFrancois Edouard A. LucasLucas Test formulatedLTefficient iterative test
187667Francois Edouard A. Lucasdisproved(?) primeLT65 iterations -- calculation was never checked
1876127Francois Edouard A. Lucasproved primeLT125 iterations
187859Fortune Landrysmallest factor foundFE143 trial divisions
188361Ivane M. Pervouchineproved primeLT59 iterations
188397H. Le Lasseursmallest factor foundFE7 trial divisions
1883151H. Le Lasseursmallest factor foundFE8 trial divisions
1883211H. Le Lasseursmallest factor foundFE4 trial divisions
1883223H. Le Lasseursmallest factor foundFE3 trial divisions
1895197Allan J. C. Cunninghamsmallest factor foundFE2 trial divisions
1908163Allan J. C. Cunninghamsmallest factor foundFE42 trial divisions
190971Allan J. C. Cunninghamsmallest factor foundFE150 trial divisions
191189R. E. Powersproved primeLT87 iterations
1911181Herbert J. Woodallsmallest factor foundFE15 trial divisions
1912173Allan J. C. Cunninghamsmallest factor foundFE? trial divisions
1914103R. E. Powersdisproved(?) primeLT101 iterations -- calculation was never checked
1914107R. E. Powersproved primeLT105 iterations
1914109R. E. Powersdisproved(?) primeLT107 iterations -- calculation was never checked
1926139Derrick H. Lehmerdisproved primeLL137 iterations
1932149Derrick H. Lehmerdisproved primeLL147 iterations
1932257Derrick H. Lehmerdisproved primeLL255 iterations
1934241R. E. Powersdisproved primeLL239 iterations
1944157Horace. S. Uhlerdisproved primeLL155 iterations
1944167Horace S. Uhlerdisproved primeLL165 iterations
1944193Horace S. Uhlerdisproved primeLL191 iterations
1945229Horace S. Uhlerdisproved primeLL227 iterations
1946-07-27199Horace S. Uhlerdisproved primeLL197 iterations
1947-06-04227Horace S. Uhlerdisproved primeLL225 iterations
1947p <= 257-Mersenne's conjecture believed to be checkedFE/LL101 & 137 were wronlgy checked or claimed by E. Fauquembergue in 1914 and 1920 to be prime
1952p <= 257 Raphael M. Robinsonchecked Mersenne numbersLLsuccessfully used first computer for Mersenne search
1952101Raphael M. Robinsondisproved primeLLshowed E. Fauquembergue miscalculated
1952137Raphael M. Robinsondisproved primeLLshowed E. Fauquembergue miscalculated
1952-01-30521Raphael M. Robinsonproved primeLLSWAC used
1952-01-30607Raphael M. Robinsonproved primeLLSWAC used
1952-06-251279Raphael M. Robinsonproved primeLLSWAC used
1953p < 2304 Raphael M. Robinsonchecked Meresenne numbersLLSWAC used
19578191David J. Wheelerdisproved primeLLIlliac I used
19572300 < p < 3300 Hans Rieselchecked Meresenne numbersLLBESK used
1957-09-083217Hans Rieselproved primeLLBESK used
19613300 < p < 5000 Alexander Hurwitzchecked Meresenne numbersLLIBM 7090 used
1961-11-034253Alexander Hurwitzproved primeLLIBM 7090 used
1961-11-034423Alexander Hurwitzproved primeLLIBM 7090 used
19625000 < p < 6000 Alexander Hurwitzchecked Meresenne numbersLLIBM 7090 used
19636000 < p < 7000 Sidney Kravitz and Murray Bergchecked Meresenne numbersLLIBM 7090 used
1983pSamuel S. Wagstaff, H. W. Lenstra, Carl B. Pomeranceconjectureguessthe number of Mersenne primes up to the limit N is about C ln ln N, with C= eg/ 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 p2 <= n.
FD Fermat's trial division
to find a factor of 2p-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 2p-1 is divisible by 2p+1
FE Euler's improved divisor theorem
If d divides 2p-1, then d = 2kp+1 for a positive integer and d = +/- 1 (mod 8)
LT Lucas 's Test
If p=4k+3, then let S0=4 else S0=3. And for i>0: Si = Si-12-2 . If and only if Sp-2 = 0 (mod 2p-1) then 2p-1 is a prime number.
LL Lucas-Lehmer Test
Let S0=4 and for i>0: Si = Si-12-2 (mod 2p-1). If and only if Sp-2 = 0 , then 2p-1 is a prime number.


Achim Flammenkamp
2006-05-04 23:12 UTC+2