CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 47 The first three pairs were given in an anonymous work.366 In 1796, J. P. Gruson100 (p. 87) gave the usual rule (1) leading to the three first known amicable pairs (verwandte Zahlen). A. M. Legendre367 attributed the rule (1) to Descartes. G. S. Kltigel368 gave a process leading to the choice of P and Q, left arbitrary by Kraft.363 We have 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, 8Q=#-7 The positive root of x2—62x— 63 = 0 lies between 60 and 61. Thus we try primes ^ 61 for R, such that R — 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 a2R2+2/3R+y under the radical sign can be made equal to the square of aR+p (p arbitrary) by choice of R. John Gough369 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 If 2^5 a/4, the first gives ayz<(l+x)a/4, while 2y-2,z>x+l by the second, Thus <?>a/4. Let a = rn, where r is a prime >1. Then <? = (a— l)/(r — 1), which with #>a/4 implies a(5-r)>4, r = 2 or 3. He proved that r?^3. whence r = 2, the case treated by van Schooten.359 J. Struve103 cited his Osterprogramm, 1815, on amicable numbers. A. M. Legendre370 discussed the amicable numbers of the type (1J of Euler364 (with Euler's m,k replaced by m— ju, #)• Legendre noted that r = 22m+A:(2fc+l)2 — 1 is of the form s2 — 1 and hence composite, if k is even; also that, if k = 3, p = 9-2m+3-l, g = 9-2m-l, one of which is of the form s2 — 1 . He considered the new case k = 7 and found f or m = 1 that p = 33023, §==257, r = 8520191, stating that if r be a prune we have the amicable numbers 2*pq, 28r. This is in fact the case.371 For & = !, we have the ancient rule (1); he proved that for n^!5 it gives only the known three pairs of amicable numbers. Paganini372, at age 16, announced the amicable numbers 1184 = 25.37, 1210 = 2.5.112, not in the list by Euler364, but gave no indication of the method of discovery. a60Encyclop6die rne'thodique . . .Amusemens des Sciences Math, et Phys., nouv. 6d., Padoue, 1793, I, 116. Cf. Lcs amusemens math., Lille, 1749, 315. l67Th6orie des nombpes, 1798, 463. >68Math. Worterbuch, 1, 1803, 246-252 [5, 1831, 55]. W9New Series of the Math. Repository (ed., Th. Leyboura), vol. 2, pt. 2, 1807, 34-39. He cited Hutton's Math. Diet., article Amicable Numbers, taken from van Schooten388. 870The"orie dea nombres, ed. 3, 1830, II, §472, p. 150. German transl. by H. Maser, Leipzig, 1893, II, p. 145. 371Tchebychef, 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. 372B. Nicold I. Paganini, Atti della R. Accad. Sc. Torino, 2, 1866-7, 362. Cf. Cremona's Ital. transl. of Baltzer's Mathematik, pt. III.