Chap. Il PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 17 
Leonard Euler^^ (1707-1783) noted that 2'' — 1 may be composite for n 
a prime; for instance, 2^' — 1 = 23-89, contrary to Wolf.'^^ If n = 4m — 1 
and 8m — 1 are primes, 2" — 1 has the factor 8m — 1, so that 2^-1 is com- 
posite for n = ll, 23, 83, 131, 179, 191, 239, etc. [Proof by Lucas.^^sj 
Furthermore, 2^^-l has the factor 223, 2^^-l the factor 431, 22^-1 the 
factor 1103, 2"^ — 1 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 2"~^(2" — 1) 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: aJ^ — V is divisible by the prime n+1, if neither a nor h is." 
[For later proofs by Euler, see Chapter III on Fermat's theorem.] Euler's 
errors as to n = 41 and 47 were corrected by Winsheim,®^ Euler^^ himself, 
and Plana.i^o 
Michael Gottlieb Hansch^ stated that 2^*— 1 is a prime if n is any of 
the twenty- two primes ^79 [error, Winsheim,^^ Kraft^^]. 
George Wolfgang Kraft^^ corrected Stifel's^^ error that 511-256 is per- 
fect and the error of Ozanam (Elementis algebrae, p. 290) that the sum of 
all the divisors of 2*" is a prime, by noting that the sum forn = 2 is 511 = 7-73 ; 
and n6ted that false perfect numbers were listed by Ozanam.'^^ Kraft 
presented (pp. 9-11) an incomplete proof, communicated to him by Tobias 
Maier [cf. Fontana^^^], that every perfect number is of Euclid's type. 
Let 1, m, n, . . .,p, A,. . .be the aliquot parts of any perfect number pA, 
where p and A are the middle factors [as 4 and 7 jn 28]. Then 
q r n m 
Solving for A, he stated that the denominator must be unity, whence 
'p = 2q/D, D = q — l—q/r — q/n — q/m. Again, D = l, whence g = 2r/D', 
D' = r — l—r/n—r/m. From I>' = 1, r = 2n/I)", D" = n — l—n/m. From 
D" = l, n = 2m/(m — 1), 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 = 2"~\ we 
get A =2" — 1. Kraft observed that if we drop from Tartaglia's^^ list of 20 
numbers those shown to be imperfect by Euler's^^ results, we have left only 
eight perfect numbers 2"-^(2"-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 WantzeP^^]. All perfect numbers end 
in 6 or 28. 
*3Comm. Acad. Petropol., 6, 1738, ad annos 1732-3, p. 103. Commentationes Arithmeticae 
Collectae, I, Petropoli, 1849, p. 2. 
^Epistola ad mathematicos de theoria arithmetices nouis a se inuentis aucta, Vindobonae 
[Vienna], 1739. 
"De numeris perfectis, Comm. Acad. Petrop., 7, 1740, ad annos 1734-5, 7-14. 
