48 History of the Theory of Numbers. (Chap, i 
P. Seelhoff^^^ treated Euler's^^ problems 1 and 2 by Euler's methods 
(though the contrary is implied), and gave about 20 pairs of amicable 
numbers due to Euler, with due credit for only three pairs. The only new- 
pairs (pp. 79, 84, 89) are 
Q2721Q in oQf83-1931 ^ef 139-863 
6 i A^-Ay'^'^|i62287 "^1167.719. 
E. Catalan^^^ stated empirically that if ??i is the sum of the divisors 
<n of n, and n2 is the sum of the divisors <ni of ?ii, etc., then n, nj, n2, . . . 
have a limit X, where X is unity or a perfect number. 
J. Perrott"^ [Perott] noted that there is no limit for n = 220, since 
^1 = 713= . . . =284, n2 = n4= . . . =220. 
H. LeLasseur^^*^ found that for n< 35 the numbers (1) are all odd primes, 
and hence give amicable numbers, only when n = 2, 4, 7. 
Josef Bezdicek^^^ gave a translation into Bohemian of Euler,^^ without 
credit to Euler, and a table of 65 pairs of amicable numbers. 
Aug. Haas"^ proved that, if M and A^ are amicable numbers, 
l/S-+l/si=l, 
m n 
where m and n range over all divisors of M and iV, respectively. For, 
2w = Sn = M+iV, so that 
m~ M ~ M ' n~ N ~ N 
If M = N, N is perfect and the result becomes that of Catalan. ^"'^ 
A. Cunningham^'^ considered the sum s{n) of the divisors <n of ?i and 
wrote s^{n) for s]s(??)}, etc. For most numbers, s^(n) = l when A; is suffi- 
ciently large. There is a small class of perfect and amicable numbers, and 
a small class of numbers n (even when n< 1000) for which s''{n) increases 
beyond the practical power of calculation [cf. Catalan^'^]. 
A. Gerardin^^" proved that the only pairs 2^-5a;, 2-yz of amicable num- 
bers, where x, y, z are odd primes, are Euler's (a), (^3) ; the only pairs 2*-23x, 
2^yz are Euler's (17), (19), (20). He cited the Exercices d'arithm^tique of 
Fitz-Patrick and Chevrel; also Dupuis' Table de logarithmes, which gives 
24 pairs of amicable numbers. 
G^rardin^^^ proved that the only pair Sxy, S2z is Euler's (60). He made 
an incomplete examination of 16-53a;, IQyz, but found no new pairs. 
3"Archiv Math. Phys., 70, 1884, 75-89. 
"*Bull. Soc. Math. France, 16, 1887-8, 129. Mathesis, 8, 1888, 130. 
'"76id., 17, 1888-9, 155-6. 
"•Lucas, Theorie dcs nombres, 1, 1891, 381. 
"'Casopis mat. a fys., Praze (Prag), 25, 1896, 129-142, 209-221. 
"8/6id., 349-350. 
"«Proc. London Math. Soc, 35, 1902-3, 40. 
""Matheshs, 6, 1906, 41^4. 
"'Sphinx-Oedipe, Nancy, 1906-7, 14-15, 53. 
