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12                           HlSTOET OF THE THEOBT  OF NtTMBEBS.                    [CHAP. I
in Waldpurg. While they considered 511X256 and 2047 X1024 as perfect, 511 has the factor 7, and (as pointed out to him by Stanislaus Pudlowski) 2047 has the factor 23. Broscius stated that
2n-l has the factor 3 5 7 11  13 17 19 23 29 31 if n is a multiple of 2 4 3  10  12    8 18 11 28    5.
The contents of the second dissertation are given below under the date 1652.
Rene* Descartes,55 in a letter to Mersenne, November 15, 1638, thought he could prove that every even perfect number is of Euclid's type, and that every odd perfect number must have the form ps2, where p is a prime. He saw no reason why an odd perfect number may not exist. For p~ 22021, s~3-7-ll-13, ps2 would be perfect if p were prime [but p = 6M92]. In a letter to Frenicle, January 9,1639, Oeuvres, 2, p. 476, he expressed his belief that an odd perfect number could be found by replacing 7, 11, 13 in s by other values.
Fermat66 stated that he possessed a method of solving all questions relating to aliquot parts. Citing this remark, Frenicle57 challenged Fermat to find a perfect number of 20 or 21 digits. Fermat58 replied that there is none with 20 or 21 digits, contrary to the opinUn of those who believe that there is a perfect number between any two consecutive powers of 10.
Fermat,59 in a letter to Mersenne, June (?), 1640, stated three propositions which he had proved not without considerable trouble and which he called the basis of the discovery of perfect numbers: if n is composite, 2n1 is composite; if n is a prime, 2n2 is divisible by 2n, and 2n 1 is divisible by no prime other than those of the form 2/cn-f-l [cf. Euler87]. For example, 211-1=23-89, 237-l has the factor 223. Also 223~1 has the factor 47, Oeuvres, 2, p. 210, letter to Frenicle, October 18, 1640.
Mersenne60 (1588-1648) stated that, of the 28 numbers* exhibited by
MDe numeris perfectis disceptatio qua ostenditur a decem millibus ad centies centena rm'llia, nullum esse perfectuni numerum atque ideo ab unitate usque ad centies centena millia quatuor tantum perfectos numerari, Amsterdam, 1638. Reproduced as the first (pp. 115-120) of two dissertations on perfect numbers, they forming pp. 111-174 of Apologia pro Aristotele & Evclide, contra Petrvm Ramvm, & alios. Addititiae sunt Dvae Discep-tationes de Nvmeris Perfectis. Authore loanne Broscio, Dantisci, 1652 (with a somewhat different title, Amsterdam, 1699).
"Oeuvres de Descartes, II, Paris, 1898, p. 429.
"Oeuvres de Fermat, 2, Paris, 1894, p. 176; letter to Mersenne, Dec. 26, 1638.
"Oeuvres de Fermat, 2, p. 185; letter to Mersenne, March, 1640.
"Oeuvres, 2, p. 194; letter to Mersenne, May (?), 1640.
"Oeuvres de Fermat, 2, pp. 198-9; Varia Opera Math. d. Petri de Fermat, Tolosae, 1679, p. 177; Precis des Oeuvres math, de P. Fermat et de T Arithme'tique de Diophante, par E. Brassinne, M6m. Ac. Imp. Sc. Toulouse, (4), 3, 1853, 149-150.
*F. Marini Mersenni minimi Cogitata Physico Mathematica, Parisiis, 1644. Praefatio Generalis, No. 19. C. Henry (Bull. Bibl. Storia Sc. Mat. e Fis., 12,1879, 524-6) believed that these remarks were taken from letters from Fermat and Frenicle, and that Mersenne had no proof. A similar opinion was expressed by W. W. Rouse Ball, Messenger Math., 21, 1892, 39 (121). On documents relating to Mersenne see rinterme*diaire des math., 2,1895, 6; 8, 1901, 105; 9, 1902, 101, 297; 10, 1903,184. Cf. Lucas.118
*0nly 24 were given by Bungus. While his table has 28 lines, one for each number of digits, there are no entry of numbers of 5, 11, 17, 23 digits.