22 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I F Landry113 soon published his table. It includes the entries (quoted by Lucas120'122): 243-1=431-9719-2099863, 247-l =-23514513-13264529, 253^1=6361-69431-20394401, 259-l = 179951-3203431780337, the least factors of the first two of which had been given by Euler.83- w This table was republished by Lucas123 (p. 239), who stated that only three entries remain in.doubt: 261-1, (261+l)/3, 264+l, each being conjectured a prime by Landry. The second was believed to be prime by Kraitchik.1130 Landry's factors of 2n+l, for 28^ rig 64 were quoted elsewhere.1136 Jules Carvallo114 announced that he had a proof that there exists no odd perfect number. Without indication of proof, he stated that an odd perfect number must be a square and that the ratio of the sum of the divisors of an odd square to itself cannot be 2. The first statement was abandoned in his published erroneous proof,133 while the second follows at once from the fact that, when p is an odd prune, the sum of the 2n-fl divisors, each odd, of p2n is odd. E. Lucas115 stated that long calculations of his indicated that 267—1 an(i 289-l are composite [cf. Cole,173 Powers186]. See Lucas20 of Ch. XVII. E. Lucas116 stated that 231 —1 and 2127 — 1 are primes. E. Catalan116 remarked that, if we admit the last statement, and note that 22-l, 23-l, 27-l are prunes, we may state empirically that, up to a certain limit, if 2n-l is a prime p, then 2P —1 is a prime g, 2a~l is a prime, etc. [cf. Catalan135]. G. de Longchamps117 suggested that the composition of 2n=|= 1 might be obtained by continued multiplications, made by simple displacements from right to left, of the primes written to the base 2. E. Lucas118 verified once only that 2127 —1, a number of 39 digits, is a prime. The method will be given in Ch. XVII, where are given various results relating indirectly to perfect numbers. He stated (p. 162) that he had the plan of a mechanism which will permit one to decide almost instantaneously whether the assertions of Mersenne and Plana that 2n-l is a prime for n = 53, 67, 127, 257 are correct. The inclusion of ?i = 53 is an error of citation. He tabulated prime factors of 2n—l for n^40. E. Lucas119 gave a table of primes with 12 to 16 digits occurring as a factor in 2n-l for n=49, 59, 65, 69, 87, and in 2n+l forn = 43, 47, 49, 53, 69, 72, 75, 86, 94, 98, 99, 135, and several even values of n> 100. The "'Decomposition des nombres 2"± 1 en leurs facteurs premiers de n = 1 a n = 64, moins quatre, Paris, 1869, 8 pp. U3aSphmx-Oedipe, 1911, 70, 95. ^L'intermediaire des math., 9, 1902, 186. 1MComptes Rendus Paris, 81,1875, 73-75. 115Sur la theorie des nombres premiers, Turin, 1876, p. 11; Theorie des nombres, 1891, 376. 116Nouv. Corresp. Math., 2, 1876, 96. U7Comptes Rendus Paris, 85, 1877, 950-2. U8Bull. Bibl. Storia So. Mat. e Fis.. 10, 1877, 152 (278-287). Lucas18-" of Ch. XVII. l"Atti R. Ac. Sc. Torino, 13, 1877-8, 279.