Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 29 
Mario Lazzarini^" attempted to prove that there is no odd perfect num- 
ber a'^b^c'^, but made the error of thinking that a is relatively prime to 
6^+. . .+6+1. He attempted to show that p = 2" — 1 is a prime if and 
only if p divides iV = 3*^+1, where fc = 2"~^ — 1 [false for a = 2, since p = S, 
N = 4:]. He restricted his argument to the case a odd, whence p = 1 (mod 3) . 
Then, if p is a prime, —3 is a quadratic residue of p, so that (— 3)^^~^^^^=1 
(mod p), whence p divides N. Conversely, when this congruence holds, he 
concluded falsely that z^=—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, 6 = 3851 are 
primes; then 
g-l 6-1 
(-3)11 + 1= _2a6,(-3) 2 =-l(mod5),(-3) ^ =(-3)^^-^^^= -1 (mod a), 
whence (—3)^^"^^^^=! (mod p). Cipolla remarked (p. 288) that we may 
deduce from a result of Lucas^^° that p is a prime if it divides N without 
dividing 3*+l for any divisor 8 of p=2"~^ — 1. 
F. N. Cole^'^ found that 2^"^ - 1 is the product of the two primes 193707721 , 
761838257287. In the footnote to p. 136, he criticized the proof by Seel- 
hoff"^ of the primality of A^ = 2^^ — 1 and stated he had verified that N is 
prime by an actual computation of a series of primes of which iV is a 
quadratic residue. 
R. D. Carmichael^^^ proved that any even perfect number Tp2\ . .p/" 
is of Euclid's type. Write d for 2'*+^ — 1. Then, as usual, 
d pf d \ p/ 
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. Cunningham^^^ gave the residues of A; =2^"*, 2*, etc., modulo 2^ — 1 for 
primes g^lOl. 
A. Turcaninov^"^ (Turtschaninov) proved that an odd perfect number 
has at least four distinct prime factors and exceeds 2000000. 
A. Gerardin^^^ noted the error by Plana. ^^° 
A. Gerardin^^^ stated the empirical laws: If n is a prime of the form 
24a^+ll and if 2" — 1 is composite, the least factor is of the form 24?/ +23 
"^Periodico di mat. insegn. sec, 18, 1903, 203; criticized by C. Ciamberlini, p. 283, and by M. 
Cipolla, p. 285. 
i"Bull. Amer. Math. Soc, 10, 1903-4, 134-7. French transl., Sphinx-Oedipe, 1910, 122-4. 
Cf. Fauquembergue.^^" 
i^^Annals of Math., (2), 8, 1906-7, 149. 
I'sproc. London Math. Soc, (2), 5, 1907, 259 [250]. 
i'6 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.is* 
'"L'interm^diaire des math., 15, 1908, 230-1. 
'"Sphinx-Oedipe, Nancy, 3, 1908-9, 113-123; Assoc, frang. 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 mUlion) is respectively 5136 y+2783, 
8016 y+335, 10896 J/+5903. 
