M E R S E N N E' S   N U M B E R S

BY RAYMOND CLARE ARCHIBALD

 T
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's texts.)
Mersenne (1588-1648), stated (in his Cogitata Physico-Mathematica, Paris, 1644, Præfatio Generalis, art. 19, folio [11]; 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] of Bungus's table. Perfect numbers are so rare that only eleven (says Mersenne) 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]. Understand this 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 authors:

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:

pCharacter of Mp
2,3,5,7,13,17,19,31,61,89,107,127Prime
11,23,29,37,41,43,47,53,59,67,71,73,79Composite and completely factored
113,151,179,223,233,239,251Incompletely factored but two or more prime factors known
83,97,131,163,173,181,191,197,211Only one prime factor known
101,103,109,137,139,149,241,257Composite but no prime factor known
157,167,193,199,227,229Character unkown

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-Œdipe, v. 3-5, 1908-10, where the cases p < 251 are referred to), and Powers (Sphinx-Œdipe, 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.

pThe Known Prime Factors of Mp First Discoverer of Factors or Character
23;
37;
531;
7127;
1123.89;Pudlowski0
138191;Codex Lat. Monac 149082, Euler3
17131071;Cataldi4, Euler3
19524287;Cataldi4, Euler3
2347.178481;Fermat1
29233.1103.2089;Euler3
312147483647;Euler3
37223.616318177;Fermat1
4113367.164511353;Plana5
43431.9719.2099863;Euler3, Landry4
472351.4513.13264529;Euler3, Reuschle7, Landry4
536361.69431.20394401;Landry4
59179951.3203431780337;Landry4
612305843009213693951;Pervouchine8, Seelhoff8, Cole9
67193707721.761838257287;Cole9
71228479.48544121.212885833; Cunningham10, Ramesam11
73439.2298041.9361973132609;Euler3, Poulet12
792687.202029703.1113491139767; Reuschle7, Lehmer13
83167.Euler3
89618970019642690137449562111; Powers14
971147.Le Lasseur15
101compositeFauquembergue16
103compositeFauquembergue16, Powers14
107162259276829213363391578010288127; Powers14, Fauquembergue16
109compositeFauqembergue16, Powers14
1133391.23279.65993.Reuschle7, Cunningham10
127170141183460469231731687303715884105727; Lucas17, Fauqembergue16
131263.Euler3
137compositeFauqembergue16
139compositeLehmer13
149compositeLehmer13
15118121.55871.165799. Le Lasseur15, Cunningham10, Kraïtchik18
157unknown
163150287.Cunningham10
167unknown
173730753.Cunningham10, Gérardin10
179359.1433.Euler3, Reuschle7
18143441.Woodall19
191383.Euler3
193unknown
1977487.Cunningham10
199unknown
21115193.Le Lasseur15
22318287.196687.Le Lasseur15, Kraïtchik18
227unknown
229unknown
2331399.135607.Reuschle7, Kraïtchik18
239479.1913.5737.176383.Euler3, Reuschle7, Bickermore20, Kraïtchik18
241compositePowers14
251503.54217.Euler3, Cunningham10
257compositeLehmer13, Kraïtchik13

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 Œdipe, 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 M113, M151, 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 Œdipe, 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 Œdipe: (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 Œdipe, 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, Récréations Mathématiques, 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 Œdipe, 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. Œ. 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 Œdipe, 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.

BROWN UNIVERSITY, PROVIDENCE, RHODE ISLAND

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