Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 23 
verification of the primality was made by H. Le Lasseur. To the latter 
is attributed (p. 283) the factorization of 2'*-! for n = 73, 79, 113. These 
had been given without reference by Lucas. ^^° 
E. Lucas^^^ proposed as a problem the proof that if 8q+7 is a prime, 
24a+3_i is not. 
E. Lucas^^^ stated as new the assertion of Euler^^ that if 4m — 1 and 
8m — 1 are primes, the latter divides A = 2*"*~^ — 1. 
E. Lucas^^^ proved the related fact that if 8m — 1 is a prime, it divides A. 
For, by Fermat's theorem, it divides 2^"*"^ — 1 and hence divides A or 
2^~^H-1. That the prime 8m — 1 divides A and not the latter, follows from 
Euler's criterion that 2^^"^^''^ — 1 is divisible by the prime p if 2 is a quad- 
ratic residue of p, which is the case if p = 8m='=l. No reference was made 
to Euler, who gave the first seven primes 4m — 1 for which 8m — 1 is a prime. 
Lucas gave the new cases 251, 359, 419, 431, 443, 491. Lucas^^^ elsewhere 
stated that the theorem results from the law of reciprocity for quadratic 
residues, again without citing Euler. Later, Lucas^^^ again expressly 
claimed the theorem as his own discovery. 
T. Pepin^^^ noted that if p is a prime and q = 2^ — 1 is a quadratic non- 
residue of a prime 4n + 1 = a^ + 6^, then qisa, prime if and only if (a — hi) / (o + hi) 
is a quadratic non-residue of q. 
A. Desboves^^^ amplified the proof by Lebesgue^^^ that every even per- 
fect number is of Euclid's type by noting that the fractional expression in 
Lebesgue's equation must be an integer which divides y^z'^ . . . and hence is 
a term of the expansion of the second member. Hence this expansion 
produces only the two terms in the left member, so that (j8+l)(7+l) . . . = 2. 
Thus one of the exponents, say /3, is unity and the others are zero. The 
same proof has been given by Lucas^^^ (pp. 234-5) and Th^orie des Nombres, 
1891, p. 375. Desboves (p. 490, exs. 31-33) stated that no odd perfect 
number is divisible by only 2 or 3 distinct primes, and that in an odd perfect 
number which is divisible by just n distinct primes the least prime is less 
than 2". 
F. J. E. Lionnet^^* amplified Euler's^^ proof about odd perfect numbers. 
F. Landry^^^ stated that 2^^=*= 1 are the only cases in doubt in his table."' 
Moret-Blanc^^° gave another proof that 2^^ — 1 is a prime. 
""Assoc. franQ. avanc. sc, 6, 1877, 165. 
»2iNouv. Corresp. Math., 3, 1877, 433. 
i"Mess. Math., 7, 1877-8, 186. AJso, Lucas."" 
i«Amer. Jour. Math., 1, 1878, 236. 
i^^BuU. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 792. The results of this paper will be cited in Ch. 
XVI. 
^Recreations math., ed. 2, 1891, 1, p. 236. 
"^Comptes Rendus Paris, 86, 1878, 307-310. 
"'Questions d'algebre 616mentau-e, ed. 2, Paris, 1878, 487-8. 
'"Nouv. Ann. Math., (2), 18, 1879, 306. 
"sBull. Bibl. Storia Sc. Mat., 13, 1880, 470, letter to C. Henry. 
"«Nouv. Ann. Math., (2), 20, 1881, 263. Quoted, with Lucas' proof, Sphinx-Oedipe, 4, 
1909, 9-12. 
