Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 47 
The first three pairs were given in an anonymous work.^^® 
In 1796, J. P. Gruson^°° (p. 87) gave the usual rule (1) leading to the 
three first known amicable pairs (verwandte Zahlen). 
A. M. Legendre^^^ attributed the rule (1) to Descartes. 
G. S. KliigeP^^ gave a process leading to the choice of P and Q, left 
arbitrary by Kraft.^^^ ^^ ^ave A:a = R+l:PQ-\-R = 2R-P-Q. Thus 
P-(-Q= \R{2A—a)—a\ /A, while PQ is given by Kraft's second equation. 
Hence P and Q are the roots of a quadratic equation. For example, if 
A = 4, then 
8P, SQ = R-7±VR^-Q2R-QS. 
The positive root of a;^ — 62a: — 63 = lies between 60 and 61. Thus we 
try primes ^ 61 for R, such that i^ — 7 is divisible by 8. The first available 
R is 71, giving P=ll, Q — 5 and the amicable pair 220, 284. In general, 
the quantity a^R^+2^R-\-y under the radical sign can be made equal to the 
square of ai^+P ip arbitrary) by choice of R. 
John Gough^^^ considered amicable numbers ax, ayz, where x, y, z are 
distinct primes not dividing a. Let q be the sum of the aliquot divisors 
of a. Then 
a+q-\-qx = ayz, x+l = {y+l){z+l). 
If q^a/i, the first gives ayz< (l+a:)a/4, while 2y-2z>x-\-l by the second, 
Thusg'>a/4. Let a = r", where r is a prime > 1 . Then Q'=(a — l)/(r' — 1), 
which with g>a/4 implies a(5— r)>4, r = 2 or 3. He proved that Tt^S. 
whence r = 2, the case treated by van Schooten.^^^ 
J. Struve^^^ cited his Osterprogramm, 1815, on amicable numbers. 
A. M. Legendre^^° discussed the amicable numbers of the type (li) of 
Euler^^^ (with Euler's m, k replaced by m—iJi,,fx). Legendre noted that 
r = 2^'""''^ (2*^+1)^ — 1 is of the form s^ — 1 and hence composite, if k is even; 
also that, if A: = 3, p = 9-2'"+3-l, g = 9-2"'-l, one of which is of the form 
s^ — 1. He considered the new case k = 7 and found for m = l that p = 33023, 
q = 257, r = 8520191, stating that if r be a prime we have the amicable num- 
bers 2^pq, 2V. This is in fact the case.^^^ For ^ = 1, we have the ancient 
rule (1); he proved that for n^l5 it gives only the known three pairs of 
amicable numbers. 
Paganini^'^^, at age 16, announced the amicable numbers 1184 = 2^37, 
1210 = 2.5.11"^, not in the list by Euler^^'*, but gave no indication of the 
method of discovery. 
^^'EncyclopMie methodique. . .Amusemens des Sciences Math, et Phys., nouv. 6d., Padoue, 
1793, I, 116. Cf. Les amusemens math., Lille, 1749, 315. 
»"Th6orie des nombres, 1798, 463. 
«8Math. Worterbuch, 1, 1803, 246-252 [5, 1831, 55]. 
«»New Series of the Math. Repository (ed., Th. Leyboum), vol. 2, pt. 2, 1807, 34-39. He cited 
Button's Math. Diet., article Amicable Numbers, taken from van Schooten^^'. 
""Theorie des nombres, ed. 3, 1830, II, §472, p. 150. German transl. by H. Maser, Leipzig, 
1893, II, p. 145. 
"iTchebychef, Jour, de Math., 16, 1851, 275; Werke, 1, 90. T. Pepin, Atti Ace. Pont. Nuovi 
Lincei, 48, 1889, 152-6. Kraitchik, Sphinx-Oedipe, 6, 1911, 92. Also by Lehmer's Fac- 
tor Table or Table of Primes. 
'"B. Nicol6 I. Paganini, Atti deUa R. Accad. Sc. Torino, 2, 1866-7, 362. Cf. Cremona's Ital. 
transl. of Baltzer's Mathematik, pt. III. 
