History of Mersenne Prime Numbers

For which prime numbers p is 2^p-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

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

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 p² <= 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 S₀=4 else S₀=3. And for i>0: S_i = S_i-1²-2 . 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 S₀=4 and for i>0: S_i = S_i-1²-2 (mod 2^p-1). If and only if S_p-2 = 0 , then 2^p-1 is a prime number.

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

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

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