Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 19 
G. W. Kraft^^ stated (p. 114) that Euler had communicated to him pri- 
vately in 1741 the fact that 2*^-1 is divisible by 2351. He stated (p. 121) 
that if 2^ — 1 is composite {p being prime), it has a factor of the form 
2q'^p + l, where g is a prime [including unity], using as illustrations the 
factorizations noted by Euler. ^^ Of the numbers 2" — 1, n a prime ^71, 
stated to be prime by Hansch,^^ six are composite, while the cases 53, . . . , 71 
are in doubt (p. 115). 
A. Saverien^^ repeated the remarks by Ens^^ without reference. 
L. Euler^^ stated in a letter to Bernoulli that he had verified that 2^^ — 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. Euler^^ gave a prime factor of 2"=»= 1 for various values of n, but no 
new cases 2^—1 with n a prime. 
L. Euler,^^ in a posthumous paper, proved that every even perfect number 
is of Euclid's type. Let o = 2"6 be perfect, where b is odd. Let B denote 
the sum of the divisors of b. The sum {2'*^^ — l)B of the divisors of a must 
equal 2a. Thus 6/5 = (2"+^-l)/2"+\ a fraction in its lowest terms. 
Hence 6 = (2^+^ — 1 )c. If c = l, 6 = 2"'*'^ — 1 must be a prime since the sum 
of its divisors is 5 = 2""^^ whence Euclid's formula. If c>l, the sum B of 
the divisors of b is not less than 6+2""''^ — 1+c+l; hence 
^^ 2"+nc+l) 2"+' 
6= h '^2"+^-l' 
contrary to the earlier equation. The proof given in another posthumous 
paper by Euler^^ is not complete. 
L. Euler^^ proved that any odd perfect number must be of the form 
y.4x+ip2^ where r is a prime of the form 4nH-l [Frenicle®^]. Express it as a 
product ABC. . . of powers of distinct primes. Denote by a, b, c, . . .the 
sums of the divisors oi A, B,C,. . ., respectively. Then abc . . . = 2 ABC .... 
Thus one of the numbers a, b, . . . , 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 =r*^"^^ In particular, no odd perfect number has the form 4n+3. 
Amplifications of this proof have been given by Lionnet,^^^ Stern, ^^'^ Syl- 
vester, ^^^ Lucas. ^" See also Liouville^° 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 [Paciuolo^®] as 
the fourteenth perfect number is imperfect; the rule by Ozanam^^ is false 
since 511 and 2047 are not primes. 
"Novi Comm. Ac. Petrop., 3, 1753, ad annos 1750-1. 
"Dictionnaire universel de math, et physique, two vols., Paris, 1753, vol. 2, p. 216. 
»^Nouv. Mim. Acad. BerUn, ann6e 1772, hist., 1774, p. 35; Euler, Comm. Arith., 1, 1849, 584. 
"Opusc. anal., 1, 1773, 242; Comm. Arith., 2, p. 8. 
"De numeris amicabihbus, Comm. Arith., 2, 1849, 630; Opera postuma, 1, 1862, 88. 
'^Tractatus de numerorum doctrina, Comm. Arith., 2, 514; Opera postuma, 1, 14-15. 
"Recreations math, et physiques par Ozanam, nouvelle 6d. par M., Paris, 1, 1778, 1790, p. 33. 
Engl, transl. by C. Hutton, London, 1803, p. 35. 
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