12 History of the Theory of Numbers. [Chap, i 
in Waldpurg. WTiile they considered 511X256 and 2047X1024 as perfect, 
511 has the factor 7, and (as pointed out to him by Stanislaus Pudlowski) 
2047 has the factor 23. Broscius stated that 
2^-1 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. 
Ren^ Descartes,^^ 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 ps^, 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, ps^ would be perfect if p were prime [but p = 61-19^]. 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. 
Fermat^^ stated that he possessed a method of solving all questions 
relating to aliquot parts. Citing this remark, Frenicle^' challenged Fermat 
to find a perfect number of 20 or 21 digits. Fermat^^ replied that there is 
none with 20 or 21 digits, contrary to the opinion of those who believe 
that there is a perfect number between any two consecutive powers of 10. 
Fermat,^^ in a letter to Mersenne, June (?), 1640, stated three proposi- 
tions which he had proved not without considerable trouble and which he 
called the basis of the discovery of perfect numbers: if n is composite, 2" — 1 
is composite; if n is a prime, 2" — 2 is divisible by 2n, and 2" — 1 is divisible 
by no prime other than those of the form 2kn-\-l [cf . Euler^']. For example, 
2"-l = 23-89, 2^^-l has the factor 223. Also 2"^^-! has the factor 47, 
Oeuvres, 2, p. 210, lett-er to Frenicle, October 18, 1640. 
Mersenne^° (1588-1648) stated that, of the 28 numbers* exhibited by 
"De numeris perfectis disceptatio qua ostenditur a decern millibus ad centies centena millia, 
nullum esse perfectum numenim 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 & Evchde, contra Petrvm Ramvm, & aUos. Addititiae sunt Dvae Discep- 
tationes de Nvmeris Perfectis. Authore loanne Broscio, Dantiaci, 1652 (with a some- 
what different title, Amsterdam, 1699). 
"Oeuvres de Descartes, II, Paris, 1898, p. 429. 
s«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. 
osQeuvres, 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 1' Arithmdtique 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 
Generahs, No. 19. C. Henry (Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 524-6) beheved 
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 Tinterm^diaire des 
math., 2, 1895, 6; 8, 1901, 105; 9, 1902, 101, 297; 10, 1903, 184. Cf. Lucas."* 
*Only 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. 
