CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 29 Mario Lazzarini172 attempted to prove that there is no odd perfect number a"6V, but made the error of thinking that a is relatively prime to 6*+. . .+6+1. He attempted to show that p=2a— 1 is a prime if and only if p divides JV = 3*+1, where fc = 2a~1 — 1 [false for a =2, since p = 3, N =4]. He restricted his argument to the case a odd, whence p=l (mod 3). Then, if p is a prime, —3 is a quadratic residue of p, so that (— 3)(p~1)/2=l (mod p), whence p divides N. Conversely, when this congruence holds, he concluded falsely that z2= — 3 (mod p) has two and only two roots, so that p is expressible in a single way as a sum of a square and the triple of a square and hence is prime. To show the error, let p — ab, where a = 23, b = 3851 are primes; then whence (-3)(p~1)/2=l (mod p). Cipolla remarked (p. 288) that we may deduce from a result of Lucas120 that p is a prime if it divides N without dividing 3a+l for any divisor 5 of p=2*~1 — 1. F. N. Cole173 found that 267 — 1 is the product of the two primes 193707721 , 761838257287. In the footnote to p. 136, he criticized the proof by Seel-hoff141 of the primality of N='2Ql-l and stated he had verified that N is prune by an actual computation of a series of primes of which AT is a quadratic residue. R. D. Carmichael174 proved that any even perfect number 2ap201. . .pn°" is of Euclid's type. Write d for 2a+1 - 1. Then, as usual, d .Pi d If n>2, pi is less than d, being an aliquot divisor of it, so that 1 + 1/p, exceeds the left member of the inequality. Hence n = 2, p2 = d. A. Cunningham175 gave the residues of fc=22n, 2*, etc., modulo 2fl— 1 for primes q ^101. A. TurSaninov176 (Turtschaninov) proved that an odd perfect number has at least four distinct prime factors and exceeds 2000000. A. G6rardin177 noted the error by Plana.110 A. Ge*rardin178 stated the empirical laws: If n is a prime of the form 24z+ll and if 2n — 1 is composite, the least factor is of the form 24t/+23 172Periodico di mat. insegn. sec., 13, 1903, 203; criticized by C. Ciamberlini, p. 283, and by M. Cipolla, p. 285. 178BuU. Amer. Math. Soc., 10, 1903-4, 134-7. French transl., Sphinx-Oedipe, 1910, 122-4. Cf. Fauquembergue.180 17< Annals of Math., (2), 8, 1906-7, 149. 17&Proc. London Math. Soc., (2), 5, 1907, 259 [250]. 17« Vest, opytn. fiziki (Spaczinskis Bote), Odessa, 1908, No. 461 (pp. 106-113), No. 463 (162-3), No. 465-6 (213-9), No. 470 (314-8). In Russian. Cf. Bourlet.168 177L'interm6diaire des math., 15, 1908, 230-1. 178Sphinx-Oedipe, Nancy, 3, 1908-9, 113-123; Assoc. franc;, avanc. sc., 1909, 145-156. In Wis- kundig Tijdschrift, 10, 1913, 61, he added that in the remaining three cases <257, n= 107, 167, 227, the least divisor (necessarily >1 million) is respectively 5136 J/+2783, 8016 y+335, 10896 J/+5903.