CHAP. I] PEKFECT, 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 2n-l for n = 73, 79, 113. These had been given without reference by Lucas.120 E. Lucas121 proposed as a problem the proof that if Sq+7 is a prime, 24fl+3-l is not. E. Lucas122 stated as new the assertion of Euler83 that if 4m— 1 and 8m — 1 are primes, the latter divides Jl=24m~"1 — 1. E. Lucas123 proved the related fact that if 8m— 1 is a prime, it divides A. For, by Fermat's theorem, it divides 28m~~2— 1 and hence divides A or 24m~~1H- 1. That the prime 8m— 1 divides A and not the latter, follows from Euler's criterion that 2(p~1)/2 — 1 is divisible by the prime p if 2 is a quadratic residue of p, which is the case if p=8m=fcl. 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. Lucas124 elsewhere stated that the theorem results from the law of reciprocity for quadratic residues, again without citing Euler. Later, Lucas126 again expressly claimed the theorem as his own discovery. T. Pepin126 noted that if p is a prime and q~2p — 1 is a quadratic non-residue of a prime 4n + 1 = &2 + &2, then q is a prime if and only if (a — hi) / (a + bi) is a quadratic non-residue of q. A. Desboves127 amplified the proof by Lebesgue107 that every even perfect number is of Euclid's type by noting that the fractional expression in Lebesgue's equation must be an integer which divides t/V . . .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 (0+ 1) (7+ 1) . . - = 2. Thus one of the exponents, say /3, is unity and the others are zero. The same proof has been given by Lucas123 (pp. 234-5) and The*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 2n. F. J. E. Lionnet128 amplified Euler's98 proof about odd perfect numbers. F. Landry129 stated that 261=fc= 1 are the only cases in doubt in his table.118 Moret-Blanc130 gave another proof that 231 — 1 is a prime. c. franc. avanc. sc., 6, 1877, 165. 121Nouv. Corresp. Math., 3, 1877, 433. 122Mess. Math., 7, 1877-8, 186. Also, Lucas.119 123Amer. Jour. Math., 1, 1878, 236. 1MBull. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 792. The results of this paper will be cited in Ch. XVI. ^Re'cre'ations math., ed. 2, 1891, 1, p. 236. 1MComptes Rendus Paris, 86, 1878, 307-310. "Questions d'algebre 616mentaire, ed. 2, Paris, 1878, 487-8. W8Nouv. Ann. Math., (2), 18, 1879, 306. "'Bull. Bibl. Storia Sc. Mat., 13, 1880, 470, letter to C. Henry. 1BONouv. Ann. Math., (2), 20, 1881, 263. Quoted, with Lucas' proof, Sphinx-Oedipe, 4, 1909, 9-12.