CHAP, i] PEKFECT, MULTIPLY PEEFECT, AND AMICABLE NUMBEBS. 13 Bungus,42 chap, 28, as perfect numbers, 20 are imperfect and only 8 are perfect: 6, 28, 496, 8128, 23550336 [for 33...], 8589869056, 137438691328, 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 l+2+22+. m m 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 1, i. e., the power 257, decreased by unity, multiplied by the power 256. [The first 11 perfect numbers are thus said to be 2n-1(2n-.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 2n—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, whatever use is made of what is already known. Mersenne61 stated that 2P —1 is a prime if p is a prune which exceeds by 3, or by a smaller number, a power of 2 with an even exponent. Thus 27-l is a prime since 7 = 22+3; again, since 67 = 3+26, 267+l = 1...7 [for 267 —1] is a prime and leads to a perfect number [error corrected by Cole173]. Understand this only of primes 2P—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,54 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 potestatern 256. 6IF. Marini Mersenni Novarvm Observationvm Physico-Mathematicarum, Tomva III, Parisiis, 1647, Cap. 21, p. 182. The Reflectiones Physico-Math. begin with p. 63; Cap. 21 is quoted in Oeuvres de Fermat, 4, 1912, pp. 67-8.