Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 13 
Bungus,^^ chap. 28, as perfect numbers, 20 are imperfect and only 8 are 
perfect : 
6, 28, 496, 8128, 23550336 [for 33. . .], 8589869056, 
137,438691328, 2305843008139952128, 
which occur at the lines marked 1, 2, 3, 4, 8, 10, 12 and 29 [for 19] of 
Bungus' table [indicating the number of digits]. Perfect numbers are so 
rare that only eleven are known, that is, three different from those of 
Bungus; norf is there any perfect number other than those eight, unless 
you should surpass the exponent 62 in 1+2+2^+ .. . The ninth perfect 
number is the power with the exponent 68 less 1; the tenth, the power 128 
less 1 ; the eleventh, the power 258 less l,i.e., the power 257, decreased by 
unity, multiplied by the power 256. [The first 11 perfect numbers are 
thus said to be 2"-'(2"-l) for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, in 
error as to n = 61, 67, 89, 107 at least.] He who would find 11 others will 
know that all analysis up to the present will have been exceeded, and will 
remember in the meantime that there is no perfect number from the power 
17000 to 32000, and no interval of powers can be assigned so great but 
that it can be given without perfect numbers. For example, if the exponent 
be 1050000, there is no larger exponent n up to 2090000 for which 2" — 1 
is a prime. One of the greatest difficulties in mathematics is to exhibit a 
prescribed number of perfect numbers; and to tell if a given number of 
15 or 20 digits is prime or not, all time would not suffice for the test, what- 
ever use is made of what is already known. 
Mersenne" stated that 2^ — 1 is a prime if p is a prime which exceeds 
by 3, or by a smaller number, a power of 2 with an even exponent. Thus 
2^-1 is a prime since 7 = 2^3; again, since 67 = 3+2*^, 2^^ + 1 = 1... 7 
[for 2®^ — 1] is a prime and leads to a perfect number [error corrected by 
Cole^^^]. Understand this only of primes 2^ — 1. Wherefore this property 
does not belong to the prime 5, but to 3, 7, 31, 127, 8191, 131071, 524287, 
2147483647, and all such. Numbers expressible as the sum or difference 
of two squares in several ways are composite, as 65 = 1+64 = 16+49. As 
he speaks of Frenicle's knowledge of numbers, at least part of his results 
are doubtless due to the latter. 
In 1652, J. Broscius (Apologia,^^ p. 121) observed that while perfect 
numbers were deduced by Euclid from geometrical progressions, they may 
be derived from arithmetical progressions: 
6 = 1+2+3, 28 = 1+2+3+4+5+6+7, 496 = 1+2+3+ ... +31. 
fNeque enim vllus est alius perfectus ab illis octo, nisi superes exponentem numerum 62, 
progressionis duplae ab 1 incipientis. Nonus enim perfectus est potestas exponentis 68, 
minus 1. Decimus, potestas exponentis 128, minus 1. Vndecimus denique, potestas 
258, minus 1, hoc est potestas 257, unitate decurtata, multiplicata per potestatem 256. 
*T. Marini Mersenni Novarvm Observationvm Physico-Mathematicarum^ Tomvs III, Parisiis, 
1647, Cap. 21, p. 182. The Reflectiones Physico-Math. begin with p. 63; Cap. 21 is 
quoted in Oeuvres de Fennat, 4, 1912, pp. 67-8. 
