CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 41 and derived as did Descartes the first three pairs of amicable numbers from 2, 8, 64. We shall see that various later writers attributed the rule to Descartes. Mersenne60 again in 1644 gave the above three pairs of amicable numbers, the misprints in both358 of the numbers of the third pair being noticed at the end of his book, and stated there are others innumerable. Mersenne61 in 1647 gave without citation of his source the rule in the form 2-2ns, 2-2nht, where * = 3-2n~l, h**2t+l, s = ht+h+t are primes [asin(l)]. Frans van Schooten,359 the younger, showed how to find amicable numbers by indeterminate analysis. Consider the pair 4#, 4yz [x, y, z odd primes]; then Eliminating x, we get 2=3-fl6/(2/— 3). The case t/ = 5 gives z- 11, z = 71, yielding 284, 220. He proved that there are none of the type 2x, 2yz, or 8x, &yz, and argued that no pair is smaller than 284, 220. For IQx, IQyz, he found 2=15+256/(t/~15), which for y=47 yields the second known pair. There are none of the type 32z, 32yz, or type 64z, 647/z. For 128$, 1287/z, he got z=127+16384/(2/-127), which for 0=191 yields the third known pair. Finally, he quoted the rule of Descartes. W. Leybourn68 stated in 1667 that " there is a fine harmony between these two numbers 220 and 284, that the aliquot parts of the one do make up the other . . . and this harmony is not to be found in many other numbers." In 1696, Ozanam73 gave in great detail the derivation of the three known pairs of "amiable" numbers by the rule as stated by Descartes, whose name was not cited. Nothing was added in the later editions.179' " Paul Halcke360 gave StifePs32 rule, as expressed by Descartes.365 E. Stone361 quoted Descartes' rule in the incorrect form that 22npq and 3-2np are amicable if p = 3-2n— 1 and # = 6-2n--l are primes. Leonard Euler362 remarked that Descartes and van Schooten found only three pairs of amicable numbers, and gave, without details, a list of 30 pairs, all included in the later paper by Euler.364 G. W. Kraft363 considered amicable numbers of the type APQ, AR, where P, Q, R are primes not dividing A. Let a be the sum of all the divisors of A. Then i Assuming prime values of P and Q such that the resulting R is prime, he sought a number A for which A /a has the derived value. For P = 3, Q = 1 1 , J68Not noticed in the correction (left in doubt) in Oeuvres de Fermat, 4, 1912, p. 250 (on pp. 66-7). One error is noted in Brosciue", Apologia, 1652, p. 154. ^"Exercitationum mathematicarum libri quinque, Ludg. Batav., 1657, liber V: sectionea triginta miscellaneas, sect. 9, 419-425. Quoted by J. Landen.88 "'Deliciae Mathematicae, oder Math. Sinnen-Confect, Hamburg, 1719, 197-9. IMNew Mathematical Dictionary, 1743 (under amicable). •"De numeria amicabiMbus, Nova Acta Eruditorum, Lipaiae, 1747, 267-9; Comm. Arith. Coll., II, 1849, 637-8. le»Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, Mem., 100-18.