22 History of the Theory of Numbers. [Chap. I 
F. Landry"^ soon published his table. It includes the entries (quoted 
byLucas^2o.i22). 
2«-l=431-9719-2099863, 2*^-1=23514513.13264529, 
253-1 = 6361.69431-20394401, 2^^-1 = 179951.3203431780337, 
the least factors of the first two of which had been given by Euler.^' •' 
This table was republished by Lucas^-^ (p. 239), who stated that only three 
entries remain in doubt: 2^^ — 1, (2^^ + l)/3, 2^*+l, each being conjectured 
a prime by Landry. The second was believed to be prime by Kraitchik.^^^" 
Landr>''s factors of 2"+l, for 28^n^64 were quoted elsewhere."^'' 
Jules Carvallo^^* announced that he had a proof that there exists no odd 
perfect number. Without indication of proof, he stated that an odd per- 
fect 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, ^^^ while the second follows at once from 
the fact that, when p is an odd prime, the sum of the 2n+l divisors, each 
odd, of p^" is odd. 
E. Lucas^^^ stated that long calculations of his indicated that 2°^ — 1 
and 2^^ - 1 are composite [cf . Cole,^" Powers^^^]. See Lucas-° of Ch. XVII. 
E. Lucas^^^ stated that 2^^ — 1 and 2^^^ — 1 are primes. 
E. Catalan^^^ remarked that, if we admit the last statement, and note 
that 2^ — 1, 2^ — 1, 2^ — 1 are primes, we may state empirically that, up to a 
certain limit, if 2" — 1 is a prime p, then 2^ — 1 is a prime g, 2^ — 1 is a prime, 
etc. [cf. Catalan^^^]. 
G. de Longchamps^^'^ suggested that the composition of 2"±1 might be 
obtained by continued multiplications, made by simple displacements from 
right to left, of the primes written to the base 2. 
E. Lucas^^^ verified once only that 2^^^ — 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 instan- 
taneously whether the assertions of Mersenne and Plana that 2" — 1 is a 
prime for n = 53, 67, 127, 257 are correct. The inclusion of n = 53 is an 
error of citation. He tabulated prime factors of 2" — 1 for n^40. 
E. Lucas^^^ gave a table of primes with 12 to 16 digits occurring as a 
factor in 2"-l for n = 49, 59, 65, 69, 87, and in 2''+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 ^ n = 64, moins quatre, 
Paris, 1869, 8 pp. 
"3<»Sphinx-0edipe, 1911, 70, 95. 
"'^L'interm^diaire des math., 9, 1902, 186. 
'"Comptes Rendus Paris, 81, 1875, 73-75. 
"'Sur la th^orie des nombres premiers, Turin, 1876, p. 11; TWorie des nombres, 1891, 376. 
"«Nouv. Corresp. Math., 2, 1876, 96. 
i^Comptes Rendus Paris, 85, 1877, 950-2. 
"sBull. Bibl. Storia Sc. Mat. e Fis.; 10, 1877, 152 (278-287). Lucas"- " of Ch. XVII. 
"'Atti R. Ac. Sc. Torino, 13, 1877-8, 279. 
