CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 25 communication from Pellet, 2n— 1 is divisible by 6n-f 1 if n and 6n+l are primes such that 6n+ 1 = 4Z/2 -\-27M2 [provided* ns= 1 (mod 4), i. e., L is odd]. M. A. Stern137 amplified Euler's98 proof concerning odd perfect numbers. E. Lucas138 repeated the statement [Desboves127] that an odd perfect number must contain at least four distinct prunes. G. Valentin139 gave a table, computed in 1872, showing factors of 2n-l for rc = 79, 113, 233, etc., but not the new cases of LeLasseur.132 The primality of AT=261 — 1, a number of 19 digits, considered composite by Mersenne and prime by Landry, was established by J. Pervusin140 and P. Seelhoff141 independently. The latter claimed to verify that there is no factor <Nl/3 of the form 8n+7, abbreviating the work by use of various numbers of which N is a quadratic residue; thus N is a prime or the product of two prunes. Since 2V = 2(230)2— 1, 2 is a quadratic residue of any priirie factor of Nj so that the factor is 8n=±= 1. It was verified that 3^=1 (mod N), where @ = (N— 1)/9. If AT=/F, where F is the prime factor 8n+l, then 3^=1 (mod F) and, by Fermat;s theorem, S^'^Umod F). It is stated without proof that one of the exponents {$ and F — l divides the other. Cole173 regarded the proof as unsatisfactory. Seelhoff proved that a perfect number of the form prrp is of Euclid's type if p and r are primes and p<r. The condition is P = If p>2, the denominator is negative. * Hence p = 2 and His statements (p. 177) about the factors of 2n — 1, n = 37, 47, 53, 59, were corrected by him (ibid., p. 320) to accord with Landry.113 P. Seelhoff142 obtained the known factors of these 2n — 1 and proved that 231 — 1 is a prime, by use of his method of quadratic residues. H. Novarese143 proved that every perfect number of Euclid's type ends in 6 or 28, and that each one > 6 is of the form 9fc+l. Jules Hudelot144 verified in 54 hours that 261 — 1 is a prime by use of the test by Lucas, Recreations math., 2, 1883, 233. "Correction by Kraitchik, Sphinx-Ocdipe, 6, 1911, 73; Pellet, 7, 1912, 15. l87Mathesis, 6, 1886, p. 248. »8/&id., p. 250. 1J8Archiv Math. Phys., (2), 4, 1886, 100-3. MOBull. Acad. Sc. St. P6tersb., (3), 31, 1887, p. 532; Melanges math. astr. ac. St. P6tersb., 6, 1881-8, 553; communicated Nov. 1883. 141Zeitschr. Math. Phys., 31, 1886, 174-8. J"Archiv Math. Phys., (2), 2, 1885, 327; 5, 1887, 221-3 (misprint for n -41). U8Jornal de sciencias math, e astr., 8, 1887, 11-14. [Servais146.] '"Matheeia, 7, 1887, 46. Sphinx-Oedipe, 1909, 16.