28 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I and considered it probable that primes of the forms 2*=*=!, 2X=*=3 (if not yielding Lucassians) generally yield prime values of 2P—1, and that no other prunes will. All known and conjectured primes 2P — 1, with p prime, fall under this rule. In a letter to Tannery,162 Lucas stated that Mersenne60'61 implied that a necessary and sufficient condition that 2P—1 be a prime is that p be a prime of one of the forms 22n-f 1, 22n=*=3, 22n+1-l. Tannery expressed his belief that the theorem was empirical and due to Frenicle, rather than to Fennat, and noted that the sufficient condition would be false if 267—1 is composite [as is the case, Fauquembergue160]. Goulard and Tannery163 made minor remarks on the subject of the last two papers. A. Cunningham164 found that 2197-1 has the factor 7487. This contradicts LeLasseur's132 statement on divisors< 30000 of Mersenne's numbers. A. Cunningham165 found 13 new cases (317, 337, 547, 937,...) in which 2q—1 is composite, and stated that for the 22 outstanding prunes q^ 257 [above list132 except 61, 197] 2«-l has no divisor < 50,000 (error as to # = 181, see Woodall184). The factors obtained in the mentioned 13 cases were found after much labor by the indirect method of Bickmore,166 who gave the factors 1913 and 5737 of 2239-l. A. Cunningham167 gave a factor of 2*-l for q = 397, 1801, 1367, 5011 and for five larger primes q. C. Bourlet168 proved that the sum of the reciprocals of all the divisors di of a perfect number n equals 2 [Catalan145], by noting that n/di ranges with di over the divisors of n, so that 2n = "Ln/di. The same proof occurs in II Pitagora, Palermo, 16, 1909-10, 6-7. M. Stuyvaert169 remarked that an odd perfect number, if it exists, is a sum of two squares since it is of the form pk2, where p is a prime 4n-fl [Frenicle, MEuler98]. T. Pepin170 proved that an odd perfect number relatively prime to 3-7, 3-5 or 3-5-7 contains at least 11,14 or 19 distinct prime factors, respectively, and can not have the form 6k+5. F. J. Studnicka171 called. Ep = 2*~l(2v-l) an Euclidean number if 2P-1 is a prime. The product of all the divisors <EP of Ep is E/"1. When Ep is written in the diadic system (base 2), it has 2p-1 digits, the first p of which are unity and the last p — 1 are zero. M2L'intenne'diaire des math., 2, 1895, 317. 163/&w*., 3,1896,115,188, 281. l«Nature, 51,1894-5, 533; Proc. Lond. Math. Soc., 26,1895, 261; Math. Quest. Educat. Times, 5,1904,108, last footnote. 165Britieh Assoc. Reports, 1895, 614. lwOn the numerical factors of an-l, Messenger Math., 25, 1895-6, 1-44; 26, 1896-7, 1-38. French transl. by Fitz-Patrick, Sphinx-Oedipe, 1912, 129-144, 155-160. U7Proc. London Math. Soc., 27,1895-6, 111. 168Nouv. Ann. Math., (3), 15, 1896, 299. "'Mathesis, (2), 6,1896, 132. 170Memoire Accad. Pont. Nuovi Lincei, 13, 1897, 345-420. l71Sitzungsber. Bohm. Gesell., Prag, 1899, math, nat., No. 30.