CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 19 G. W. Kraft93 stated (p. 114) that Euler had communicated to him privately in 1741 the fact that 247- 1 is divisible by 2351. He stated .(p. 121) that if 2P — 1 is coinposite (p being prime), it has a factor of the form 2gmp+l, where q is a prime [including unity], using as illustrations the factorizations noted by Euler.83 Of the numbers 2n— 1, n a prime ^71, stated to be prime by Hansch,84 six are composite, while the cases 53, . . . , 71 are in doubt (p. 115). A. Saverien94 repeated the remarks by Ens51 without reference. L. Euler95 stated in a letter to Bernoulli that he had verified that 231 — 1 is a prime by examining the primes up to 46339 which are contained in the possible forms 248n+l and 248n+63 of divisors. L. Euler96 gave a- prime factor of 2n=*=l for various values of n, but no new cases 2n — 1 with n a prime. L. Euler,97 in a posthumous paper, proved that every even perfect number is of Euclid's type. Let a=2nb be perfect, where b is odd. Let B denote the sum of the divisors of b. The sum (2n4~1 — 1)J5 of the divisors of a must equal 2a. Thus b/J?=--(2n+1-l)/2n'H, a fraction in its lowest terms. Hence 6 = (2n+1 — l)c. If c = l, &=2n"H — 1 must be a prime since the sum of its divisors is B=2n+l, whence Euclid's formula. If c> 1, the sum B of the divisors of b is not less than &+2n+1 — 1+6+1; hence ^ 2ft+1 6= b >2»+1-l' contrary to the earlier equation. The proof given in another posthumous paper by Euler98 is not complete. L. Euler98 proved that any odd perfect number must be of the form r4X+1P2, where r is a prime of the form 4n+l [Frenicle64]. Express it as a product ABC ... of powers of distinct primes. Denote by a, 6, c, . . . the sums of the divisors of A, B, C, . . . , respectively. Then abc . . . = 2 ABC .... Thus one of the numbers a, &, . . . , say a, is the double of an odd number, and the remaining ones are odd. Thus B, C, . . .are even powers of primes, while A = r4X+1. In particular, no odd perfect number has the form 4n+3. Amplifications of this proof have been given by Lionnet,128 Stern,137 Sylvester,149 Lucas.187 See also Liouville30 in Chapter X. Montucla" remarked that Euclid's rule does not give as many perfect numbers as believed by various writers; the one often cited [Paciuolo16] as the fourteenth perfect number is imperfect; the rule by Ozanam79 is false since 511 and 2047 are not primes. MNovi Comra. Ac. Pctrop., 3, 1753, ad annos 1750-1. MDictionnaire universe! dc math, et physique, two vols., Paris, 1753, vol. 2, p. 216. wNouv. M6m. Acad. Berlin, ann6e 1772, hist., 1774, p. 35; Euler, Comm. Arith., 1, 1849, 584. B60pusc. anal., 1, 1773, 242; Comm. Arith., 2, p. 8. 97De numeris arnicabilibus, Comm. Arith., 2, 1849, 630; Opera postuma, 1, 1862, 88. 98Tractatus de numerorurn doctrina, Comm. Arith., 2, 514; Opera posturna, 1, 14-15. "Recreations math, et physiques par Ozanam, nouvelle id. par M., Paris, 1, 1778, 1790, p. 33. Engl. transl. by C. Button, London, 1803, p. 35.