M E R S E N N E' S N U M B E R S|
BY RAYMOND CLARE ARCHIBALD
HE 55 numbers here in
question are Mp =
2p - 1,
p (prime) = 2, 3,..., 257. They were considered in discussion
of perfect numbers, that is, numbers equal to the sum of their divisors
other than themselves. Euclid showed, in effect (Elements, book 9,
prop. 36), that a formula for such numbers is
2p - 1
(2p - 1), if
2p - 1 ( = 1 + 2 +
2 2 + ... +
2p - 1) is a prime. Only 12
values of p are known to make
2p - 1 prime.
These values yield the twelve known perfect numbers. The tenth perfect
number (in the order of discovery) was found as recently as 1876
(p = 127), the eleventh in 1911 (p = 89) and the twelfth
in 1914 (p = 107). The first four perfect numbers (6, 28, 496,
8128) were known to Nicomachus (c. 100). A fifth was added in
the fifteenth century, and the sixth and seventh were derived in the
sixteenth. At the end of the sixteenth century an author of many
surmises in this regard was Peter Bungus (d. 1601), in his work of
1584. (For details here and later see Dickson, Hist. of the Theory
of Numbers, v. 1, 1919; in the next paragraph about Mersenne I
quote the remarkably accurate statements of Dickson, which I checked with
Mersenne (1588-1648), stated (in his Cogitata
Physico-Mathematica, Paris, 1644, Præfatio Generalis, art.
19, folio ; this Latin passage is reprinted in W. W. R. Ball,
Mathematical Recreations and Essays, fifth ed., London, 1911,
p. 333-334; and also in Amer. Journ. Math., v. 1, 1878,
p. 235) that
of the 28 [mistake for 24]
numbers exhibited by Bungus,
in chap. 28 of his Numerorum Mysteria, as perfect numbers
20 are imperfect and only 8 are perfect:
6, 28, 496, 8128, 23550336 [for 33 ..., misprint of Mersenne],
8589869056, 137438691328, 2305843008139952128,
which occur at lines marked 1, 2, 3, 4, 8, 10, 12, and 29 [for 19]
Bungus's table. Perfect numbers are so rare that only eleven (says
are known, that is, three different from those of Bungus;
nor is there any perfect number other than those eight, unless you
would surpass the exponent 62 in 1 + 2 + 22 + ....
The ninth perfect number is the power with the exponent 68 less 1; the tenth,
the power 128 less 1; the eleventh, the power 258 less 1, i. e., the
power 257, decreased by unity multiplied by the power 256. [The first
11 perfect numbers are thus said to be
2p - 1
(2p - 1) for p =
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, in error as to 61, 67, 89, 107, 257;
there's a possibility of six more errors.]
He who would find 11 others will
know that all analysis up to the present will have been exceeded, and will
remember in the meantime that there is no perfect number from the power 17000
to 32000, and no interval of powers can be assigned so great but that
it can be given without perfect numbers. For example, if the exponent
be 1050000, there is no larger exponent p up to 2090000 for
which 2p - 1 is a prime.
One of the greatest difficulties in mathematics is to exhibit a prescribed
number of perfect numbers, and to tell if a given number of 15 or 20 digits
is prime or not; all time would not suffice for the test, whatever use is
made of what is already known.
Mersenne stated also (in his "Reflectiones
physico-mathematicae," chapter 21, p. 182, of his Novarvm
Observationvm Physico Mathematicarvm, Tomus 3, Paris, 1647; this
work was called Tomus 3 since the Cogitata was regarded as v. 1,
and Universae Geometriae, Mixtaequae Mathematicae Synopsis, Paris,
1644, as v. 2) that
2p - 1
is a prime if p is a prime which exceeds by 3, or by a smaller number,
a power of 2 with an even exponent. Thus 27 - 1
is a prime since 7 = 22 + 3; again,
since 67 = 3 + 26,
267 + 1 = 1 ... 7
[for 267 - 1]
is a prime and leads to a perfect number [not so].
only of primes 2p - 1.
Wherefore this property does not belong to the prime 5, but to 3, 7, 31,
127, 8191, 131071, 524287, 2147483647, and all such.
Having now before us precisely what Mersenne stated
in this connection we see exactly how accurate it is to affirm that
Mersenne asserted, that the only values of p not greater than
257 which make Mp prime are
p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. In view of our
second reference to Mersenne, Ball's assumption that 67 was a misprint
for 61 is absurd.
It is the purpose of this paper to give accurate and
complete information concerning present knowledge of
Mp and to indicate the first author
of each result, with a bibliographic reference. If factors of an
Mp are given, I indicate
who found each factor, but I make no reference to an earlier author who
may have shown this Mp composite,
without finding any factor. It will be noted that no proved result is
attributed to Mersenne, and only slight direct results to his contemporary
Fermat. The chief tabular results previously given were by the following
L. Euler, Opuscula Varii Argumenti, v. 2,
Berlin, 1750, p. 27; table of the prime factors of
2p - 1,
p = 2, ..., 36.
K. F. Reuschle, Mathematische Abhandlungen . . .
Neue zahlen-theoretische Tabellen, Stuttgart, 1856, p. 21, 42-53.
F. Landry, (a) Décomposition des Nombres
2n ± 1 en leurs Facteurs
Premiers de n = 1 à n = 64, moins quatre, Paris,
1869, p. 6-7; (b) Amer. Journ. Math., v. 1, 1878, p. 239-240.
Le Lasseur de Sanzey, Bulletino d. Bibliografia
e d. Storia (Boncompagni), v. 11, Dec. 1878, p. 788-789.
W. W. R. Ball, Mathematical Recreations and Essays,
fifth ed., London, 1911, chapter 15, "Mersenne's numbers;" table, p. 336.
H. J. Woodall and A. J. C. Cunningham, Manchester
Lit. and Phil. So., Memoirs and Proc., v. 56, no. 1, Apr. 1912, p. 2-3.
A. J. C. Cunningham, Proc. Fifth Intern. Congress
Mathems. Cambridge, 1913, v. 1, p. 384-386.
A. J. C. Cunningham and H. J. Woodall, (a)
Factorisations of (yn ± 1),
London, 1925, p. xiv-xv, 1-3; (b) Haupt-Exponents, Residue-Indices, Primitive Roots,
and Standard Congruences, London, 1922, p. 1-30, 101-131; T. G.
Creak was a joint author of this latter volume.
M. Kraïtchik, (a) Théorie des Nombres, v. 1,
Paris, 1922, p. 218; (b) Recherches sur la Théorie des Nombres,
Paris, v. 1, 1924, p. 24, v. 2, 1929, p. 84; (c) L'Échiquier,
v. 3, Oct. 1927, p. 756. See also note 18 below.
D. H. Lehmer, (a) Sphinx, v. 1, Nov. 1931,
p. 164-165; (b) Amer. Math. So., Bull.,
v. 38, June 1932, p. 384.
In the light of present knowledge there are
omissions or errors of statement, in connection with
Mp, in most of these tables.
Misstatements, in view of available information at the time the tables
were published, are to be found even in some tables of Cunningham and
Kraïtchik. In corrected form the table of Lehmer (a revision of
a similiar table of Cunningham and Woodall, (a) p. xiii) indicating the
character of each of the Mp,
is as follows:
|p||Character of Mp|
but two or more prime factors known|
|83,97,131,163,173,181,191,197,211||Only one prime factor
|101,103,109,137,139,149,241,257||Composite but no prime
Hence we have complete information concerning only
25, of 55, Mp. It has been shown by
Cunningham and Gérardin (see Proc. Fifth Intern. Congress Mathems.,
v. 1, 1913, p. 384-385; Br. Assoc. Adv. Sci., Report, 1911,
p. 321; Sphinx-dipe, v. 3-5, 1908-10, where the
cases p < 251 are referred to), and Powers
(Sphinx-dipe, v. 8, Apr. 1913, p. 49-50, for
the case p = 257) that no one of these last 14
Mp can have a factor
< 106. We now present the
details in a table with notes, more complete than any previously
published. In order to avoid type-setting difficulties, information
normally given in footnotes is collected in the text after
the table. Following Cunningham a semicolon (;) indicates that the
factorization is complete for the value of p in question. It
will be seen that factorization is complete through p = 79.
A dot (.) separates factors and, following the last factor,
indicates that further prime factors are unknown.
|p||The Known Prime Factors of
||First Discoverer of Factors or Character|
|13||8191;||Codex Lat. Monac 149082,
Le Lasseur15, Cunningham10,
0 Stanislaus Pudlowski to whom
the result is attributed by J. Brozek or Broscius in his De Numeris
Perfectis, Amsterdam, 1638; and by J. Caramuel in his Mathesis
Biceps, Campania, 1670, v. 1, p. 46.
1 Fermat indicated the
factors of Mp, for p =
11, 23, 37 in a letter of 18 Oct. 1640; see Oeuvres de Fermat, v. 2,
1894, p. 209-210.
2 This codex dated
1461 gives the fifth perfect number correctly (see Bibliotheca
Mathematica, s. 2, v. 9, 1895, p. 41). Moreover the
work seems to suggest an understanding derivation of the result.
3 Euler gave
a table of the prime factors of
2p - 1 for p =
2,..., 37, in his Opuscula Varii Argumenti, v. 2, Berlin,
p. 27, G. W. Kraft stated (Novi Comm. Acad. Petrop., v. 3,
1753, p. 114) that Euler had communicated to him privately in 1741 that
247 - 1 is divisible by 2351. Euler
gave the smallest factors for p = 29, 43, 73, "etc.," in 1732,
Comment. Acad. Sci. Petrop., v. 6, 1738, p. 10; Euler,
Opera Omnia, s. 1, v. 2, 1915, p. 3. He stated that
it followed from his general theorem (proved by Lagrange in 1775) that if
p = 4m - 1, and 8m - 1 are primes,
2p - 1 is divisible by
8m - 1. In the case of p = 83 this
gives the factor 167; of p = 131, the factor 263; of
p = 179, the factor 359; of p = 191, the factor 383;
of p = 239 the factor 479; and of p = 251 of the factor
503. Since for p = 31, 8m - 1 is not a
prime this theorem does not apply, but Euler settled this case also
in Nouv. Mem. d. l'Acad. d. Sc. de Berlin 1772, 1774, Histoire,
p. 36; Euler, Opera Omnia, s. 1, v. 3, 1917, p. 336-337.
See also Legendre, Théorie des Nombres, third ed., Paris,
1830, v. 1, p. 228-229.
4 P. A. Cataldi (1548-1626), Trattato
de Numeri Perfetti, Bologna, 1603 [composed in 1588, according
to the preface], p. 12-22, giving the proof that Mp
are primes, for p = 13, 17, 19, by trying as possible divisors
every prime less than their respective square roots. Thus the first
seven perfect numbers were known at this time. Bungus and others
earlier do not seem deserving of credit in this connection since their
statements appear to be guess-work. Mersenne stated the eighth perfect
number correctly, but it took an Euler to give the proof.
5 G. A. A. Plana, R. Accad. d. Sc., Turin,
Memorie, s. 2, v. 20, 1863, p. 130, in memoir presented 26 June 1859.
6 F. Landry, Décomposition des
Nombres 2n ± 1 en leurs Facteurs
Premiers, Paris, 1869, p. 7 the factors for p = 53, the third factor
for p = 47, and the second and third factors for p = 43.
These results seem to have been known earlier; see F. Landry,
Aux. Mathématiciens de toutes les parties du monde;
communications sur la décomposition des nombres en leurs
facteurs simples, Paris, 1867, p. 8. For p = 59, see
F. Landry, Amer. Journ. Math., v. 1, 1878 p. 240.
7 K. G. Reuschle, Mathematische
Abhandlungen . . . Neue zahlen-theoretishce Tabellen, Stuttgart,
1856, p. 42-53; gave the factor 1913 for p = 239; the smaller
factor for p = 233; 3391 for p = 113; the smallest
factor for p = 79, and the factor 4513 for p = 47.
8 P. Seelhoff, Zeitsch. f. Mathem.
u. Physik, v. 31, 1886, p. 177-178. But I. M. Pervouchine found in
1883 the result that M61 is prime. This is set
forth in Bull. Acad. d. Sci. St. Pétersburg, s. 3, v. 31,
1887, cols. 532-533; and Mélanges Mathém. et Astron.
tirés du Bull. de l'Acad. . . . d. Sci. de St. Pétersbourg,
v. 6 (1881-1888), p. 553-554, where there is a statement from a report
by Imchenetzki and Bouniakowsky. Pervouchine's paper on the subject
was presented to the Academy 20 Dec. 1883 and a definite statement
was made as to its contents in the "Memoires Russes de l'Académie
(Zapiski Imperatorskoi Akademii)," v. 48. I have not
been able to identify this last publication.
9 F. N. Cole, Amer. Math. So., Bull.,
v. 10, Dec. 1903, p. 134-137. Cole here notes that he had given a
rigorous proof that Mp was prime for
p = 61, and that the tests of Seelhoff were insufficient.
10 A. J. C. Cunningham, Proceedings
of the Fifth International Congress of Mathematicians,
Cambridge, v. 1, 1913, p. 385: for p = 71, the smallest factor
(L'Interméd. d. Mathém., v. 16, 1909, p. 252; see also
Nature, v. 81, 12 Aug. 1909, p. 194); for p = 113 the
two larger factors are given and it is stated that they were found
in 1908-09; similiarly that the second factor for p = 151 was
found in 1909; the factor for p = 163 in 1908 (also Nature,
v. 78, 30 Apr. 1908, p. 23; London Math. So., Proc., s. 2, v. 6,
p. xxii, meeting 30 Apr. 1908); the factor for p = 173 (also
Sphinx dipe, v. 7, Feb. 1912, p. 38 -this result was found
in collaboration with A. Gérardin); the factor for p = 197
(also Nature, v. 51, 4 Apr. 1895, p. 533; London Math So.,
Proc., v. 26, p. 261, meeting 14 Mar. 1895); for p = 251
the larger factor, found in 1909. The above mentioned factors of
M251 were also announced by
Cunningham in Manchester Lit. and Phil. So., Mem. and Proc.,
v. 56, no. 1, Apr. 1912, p. 3.
11 V. Ramesam, Indian
Math. So., Journ., v. 4, Apr. 1912, p. 56, giving
the second and third factors not found by Cunningham; also in Nature,
v. 89, 28 Mar. 1912, p. 80.
12 P. Poulet,
Sphinx dipe, v. 18, Apr. 1923, p. 64, gave
the two larger factors.
13 D. H. Lehmer, Amer. Math. So.,
Bull.: (a) v. 39, Feb. 1933, p. 108, giving the second and
third factors for p = 79; (b) v. 32, Sept.-Oct. 1926, p. 522,
p = 139 composite; (c) v. 38,
June 1932, p. 383-384, showing
Mp composite for p = 149,
and p = 257; also Sphinx, v. 1, May 1931,
p. 31-32 (p = 257) and Nov. 1931, p. 163-165
(p = 149). In 1922, M. Kraïtchik arrived
at a similiar result for p = 257 but was unwilling to guarantee
the accuracy of his conclusion; cf. M. Kraïtchik, Recherches
sur la Théorie des Nombres, Paris, v. 1, 1924, p. 21, and
Théorie des Nombres, v. 2, 1926, p. 142.
It may by pointed out that the factors for p = 79
were found by Dr. Lehmer by means of a machine which he constructed
with the aid of a grant from the Carnegie Institution of Washington,
and advice of Dr. R. C. Burt of Pasadena. See (1) Amer. Math. So.,
Bull., v. 38, Sept. 1932, p. 635; (2) "A photo-electric number
sieve," Amer. Math. Mo., v. 40, 1933, p. 401-406; (3) "A
machine for combining sets of linear congruences," Mathem.
Annalen, v. 109, May 1934, p. 661-667.
In 1875-77 E. Lucas made claims, never later proved,
concerning a calculating machine. Passages in his writings referring
to this are as follows: (1) "J'ai trouvé le plan d'un mécanisme
assez simple, qui permettra de vérifier, automatiquement et en
très peu de temps, les assertions du P. Mersenne, et de trouver
de très grands nombres premiers de 80 et même de 100 chiffres
compris dans la forme an ± 1,
a étant égal à 2, 3, ou 5. La construction de ce
mécanisme permet de calculer rapidment, dans le système binaire
de la numération, les résidus des vn
par rapport au nombre dont on cherche la décomposition en facteurs premiers,
et repose, d'une part, sur les théorèmes qui précèdent,
et d'autre part, sur les lois mathématiques de la géométrie
du tissage." (Institut de France, Acad. d. Sc., Comptes Rendus, v. 82,
5 June 1875, p. 1305); (2) "J'ai conçu, en suivant cette voie, le plan
d'un mécanisme qui permettrait de décider du mode de composition
de ces nombres, et de trouver des nombres premiers ayant mille
chiffres, dans le système décimal, et même beaucoup plus."
(Assoc. Fr. l'Avanc. d. Sci., Comptes Rendus, 1876, p. 68); (3)
"Je ferai seulement observer, pour l'instant, que j'ai trouvé le
plan d'un mécanisme qui permettra de décider presque
instantanément si les assertions du Père Mersenne et du Baron
Plana, rapportées dans cette Note sur les nombres
253 - 1, 267 - 1,
2127 - 1, 2257 - 1,
qu'ils considéraient comme premiers, sont exactes." [Bull. d.
Bibl. e d. Storia (Boncompagni), v. 10, 1877, p. 162].
A. Gérardin published the following statement in
his Sphinx dipe: (a), v. 17, Apr. 1922, p. 64, "J'ai
terminé, après de longues recherches, la mise au point de
la première machine d calculs binaires, généralisation
des travaux et des projets d'Éd. Lucas. Mon modèle d'études
et en construction. La machine permet la décision rapide pour les
Nombres de Mersenne, de Fermat et d'autres formes de nombres cités
par Ed. Lucas (primalité, puis factorisation)." (b), v. 18,
Feb. 1923, p. 18, for p = 157, 229, 241, "S'ils
répondent à ma loi empirique envisigée, on aurait
des nombres composés. J'espère en donner, cette
année, une démonstration, directe et rapide, à
l'aide de ma nouvelle machine à calcul binaires, mise au point
depuis longtemps." I find no record that this machine was ever completed.
It seems strange that he makes no reference to the "Piano
arithmétique pour la vérification des grands nombres premiers"
exhibited in 1891 by Genaille; this was for testing the primality of
Mp (see Assoc. Fr. l'Avanc. d .Sci. Comptes
Rendus, part 1, p. 159). Note also the machine of T. R. Mason,
Amer. Math. So., Bull., v. 21, p. 68, Nov. 1914, and Indiana
Acad. Sci., Proc., 1914, p. 429. The object of the machine
of Lucas was simply to test for primality numbers of the Fermat and
Mersenne type, by applying his sufficient conditions for primality
in a mechanical way. When completed the machine would perform
multiplication and division by large numbers written to the base 2.
But with such a machine one cannot factor a number which tests out
to be composite. [In writing this note I am indebted to Dr. D. H.
Lehmer for several suggestions.]
14 R. E.
Powers:for p = 89,
Amer. Math. Mo., v. 18,
Nov. 1911, p. 195-197; for p = 103 and 109,
London Math. So., Proc., s. 2, v. 15, p. xxii,
announcements at a meeting 10 Feb. 1916 of calculations completed in
June 1914; for p = 107, Sphinx dipe,
v. 9, June 1914, p. 105-108; also Amer. Math. So.,
Bull., v. 20, July 1914, p. 531; also
London Math. So., Proc., s. 2,
v. 13, p. xxxix; for p = 241, Amer. Math. So.
Bull., v. 40, Dec. 1934, p. 383. As another instance
of the unreliability of a statement of Lucas, note his affirmation that "after
long calculations," he found that M89
is composite (Lucas, Théorie des Nombres, v. 1, Paris,
1891, p. 376).
15 É. Lucas,
v. 2, 1883, p. 230; for the case p = 151,
Le Lasseur de Sanzey is credited with only the first factor.
16 E. Fauquembergue,
Sphinx dipe, v. 9, June 1914, p. 85, 103-105
giving details of his work for p = 101, 103, 107, 109, 127.
The work of Powers for p = 107 is printed in the same
issue (p. 105-108). The checking of Lucas's work for p = 127
was important. Fauquembergue's result for p = 107 was
received by the editor of S. . 7 June 1914.
Power's cable announcing this same result was sent to the
London Math. So. on 1 June 1914. For p = 137, see
Sphinx dipe, v. 15, Feb. 1920, p. 17-18.
17 É. Lucas:
(a) Institut de France, Acad. d. Sci., Comptes Rendus, v. 82,
10 Jan. 1876, p. 167; after stating certain theorems he asserts:
"C'est à l'aide de ces Théorèmes que je pense avoir
démontré que le nombre
A = 2127 - 1 est
premier." (b) Bulletino d. Bibliografia e d. Storia (Boncompagni),
v. 10, Apr. 1877, p. 152; he states here, "J'ai ainsi
verifié mais une seule fois, je l'avoue, que le nombre
A = 2127-1 est
une nombre premier; de nombre contient trente-neuf chiffres." The
correctness of this statement was checked in 1914 by Fauquembergue.
16 This is the largest known prime.
18 In M. Kraïtchik,
Recherches sur la Théorie des Nombres, Paris, 1924,
p. 24, 165, 175, 170, I found the fourth factor for p = 239,
the second for p = 233, the second for p = 223,
and the third for p = 151. He gave these results earlier,
however, on the sixth photographic sheet accompanying the 8 pages of text
entitled, Tables de la plus grande solution de la Congruence
2(p - 1) / x =
1 (mod. p) pour tous les nombres premiers p inférieurs
à 300000 exepté les cas de x = 1 ou 2, and published at
Nancy in Aug. 1921.
19 H. J. Woodall,
London Math. So., Proc., s. 2, v. 9, 1911, p. xvi,
result announced at the meeting 8 June 1911; also Manchester Lit.
and Phil. So., Memoirs and Proc., v. 56, no. 1, Apr. 1912,
p. 4; Woodall gives the number of 50 digits, which results from dividing
M181 by 43441. On page 5
Woodall gives the "forms of the only possible divisors of incompleted
Mersenne's numbers." For example, for p = 83,
664n + 1 and 664n + 167; and for
p = 257, 2056n + 1, and
2056n + 1543. These divisors through p = 101
were given by Plana, l.c., p. 137.
20 C. E. Bickmore,
Messenger of Math., v. 25, May 1895, p. 19, giving
the factor 5737.