CHAP, il PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 17 Leonard Euler83 (1707-1783) noted that 2n— 1 may be composite for n a prime; for instance, 2n- 1=23 -89, contrary to Wolf.77 If n = 4m— 1 and Sin— I are primes, 2n— 1 has the factor 8m— 1, so that 2n — 1 is composite for n = ll, 23, 83, 131, 179, 191, 239, etc. [Proof by Lucas.123] Furthermore, 237-l has the factor 223, 243-l the factor 431, 229-l the factor 1103, 273-l the factor 439, etc. "However, I venture to assert that aside from the cases noted, every prime less than 50, and indeed than 100, makes 2n~1(2n~l) a perfect number, whence the eleven values 1, 2, 3, 5, 7, 13, 17, 19, 31, 41, 47 of n yield perfect numbers. I derived these results from the elegant theorem, of whose truth I am certain, although I have no proof: an—bn is divisible by the prime n+1, if neither a nor 6 is." [For later proofs by Euler, see Chapter III on Fennat's theorem.] Euler's errors as to n=41 and 47 were corrected by Winsheim,90 Euler93 himself, and Plana.110 Michael Gottlieb Hansch84 stated that 2n— 1 is a prime if n is any of the twenty-two primes ^79 [error, Winsheim,90 Kraft93]. George Wolfgang Kraft85 corrected Stifel's31 error that 511-256 is perfect and the error of Ozanam (Elementis algebrae, p. 290) that the sum of all the divisors of 24n is a prime, by noting that the sum f or n = 2 is 51 1 = 7-73 ; and noted that false perfect numbers were listed by Ozanam.79 Kraft presented (pp. 9-11) an incomplete proof, communicated to him by Tobias Maier [cf. Fontana101], that every perfect number is of Euclid's type. Let 1, m, ft,. . .,p, A,. . .be the aliquot parts of any perfect number pA, where p and A are the middle factors [as 4 and 7 in 28]. Then r n m Solving for A, he stated that the denominator must be unity, whence p = 2q/D, D = q—I—q/r--q/n~q/m. Again, D = l, whence q = 2r/t)', D'=r-l-r/n-r/w. From D' = l, r = 2n/£", D" = n-l-n/m. From 2)* = !, n = 2m/(w-l), m — 1 = 1, m = 2, n = 4, r = 8, etc. Thus the aliquot parts up to the middle must be the successive powers of 2, and A must be a prime, since otherwise there would be new divisors. For p = 2n~1, we get A =2n-l. Kraft observed that if we drop from Tartaglia's38 list of 20 numbers those shown to be imperfect by Euler 's83 results, we have left only eight perfect numbers 2n~1(2n-l) for n^39, viz., those for n = 2, 3, 5, 7, 13, 17, 19, 31. For these, other than the first, as well as for the false ones of Tartaglia, if we add the digits, then add the digits of that sum, etc., we finally get unity (p. 14) [proof by Wantzel106]. All perfect numbers end in 6 or 28. 83Comm. Acad. Petropol., 6, 1738, ad annos 1732-3, p. 103. Commentationea Arithmeticae Collectae, I, Petropoli, 1849, p. 2. ^Epietola ad mathematicos de theoria arithmetices nouis a se inuentis aucta, Vindobonae [Vienna], 1739. MDe numeris nerfectis. Comm. Acad. Petroi).. 7. 1740. ad annos 1734-5. 7-14.