40 HlSTOBY OF THE THEORY OF NUMBERS. [CHAP. I Alkalacadi,352 a Spanish Arab (11486), showed the method of finding the least amicable numbers 220, 284. Nicolas Chuquet18 in 1484 and de la Roche28 in 1538 cited the amicable numbers 220,284," de merueilleuse familiarite lung auec laultre." In 1553, Michael Stifel32 (folios 26v~27v) mentioned only this pair of amicable numbers. The same is true of Cardan,27 of Peter Bungus42 (Mysticae numerorum signif., 1585, 105), and of Tartaglia.353 Reference may be made also to Schwenter.62 In 1634 Mersenne301 (p. 212) remarked that "220 and 284 can signify the perfect friendship of two persons since the sum of the aliquot parts of 220 is 284 and conversely, as if these two numbers were only the same thing." According to Mersenne's304 statement in 1636, Fermat354 found the second pair of amicable numbers 17296=24-2347, 18416=2M 151, and communicated to Mersenne305 the general rule: Begin with the geometric progression 2, 4, 8,..., write the products by 3 in the line below; subtract 1 from 5 11 23 47 the products and enter in the top row. The 24 8 16 bottom row is 642-1, 12-24-1,.. .When a 6 12 24 48 number of the last row is a prune (as 71) and 71 287 1151 the one (11) above it in the top row is a prime, and the one (5) preceding that is also a prime, then 71.4=284, 5-11-4 = 220 are amicable. Similarly for 1151-16 = 18416, 2347-16 = 17296, and so to infinity. [The rule leads to the pair 2nhtt 2ns, where h, t, s are given by (1).] Descartes355 gave the rule: Take (2 or) any power of 2 such that its triple less 1, its sextuple less 1, and the 18-fold of its square less 1 are all primes;* the product of the last prime by the double of the assumed power of 2 is one of a pair of amicable numbers. Starting with the powers 2, 8, 64, we get 284, 18416, 9437056, whose aliquot parts make 220, etc. Thus the third pair is 9363584 = 2M91-383, 9437056 = 27-73727. Descartes356 stated that Fermat's rule agrees exactly with his own. Although we saw that Mersenne quoted in 1637 the rule in Fermat's form and expressly attributed it to Fermat, curiously enough Mersenne314 gave in 1639 the rule in Descartes' form, attributing it to "un excellent Ge*ometre" (meaning without doubt Descartes, according to C. Henry357), "'Manuscript in Bibliotheque Nationals Paris, a commentary on the arithmetic Talkhys of Ibn Albanna (13th cent.). Cf. E. Lucas, L'arithme'tique amusante, Paris, 1895, p. 64. "'Quesiti et Inventione, 1554, fol. 98 v. ^Oeuvres de Fermat, 2,1894, p. 72, letter to Roberval, Sept. 22,1636; p. 208, letter to Frenicle, Oct. 18, 1640. WQeuvres de Descartes, 2,1898, 93-94, letter to Mersenne, Mar. 31,1638. *Evidently the numbers (1) if the initial power of 2 be 2n~1. ^Oeuvres de Descartes, 2, 1898, 148, letter to Mersenne, May 27, 1638. '"Bull. Bibl. Storia Sc. Mat. e Fis.3 12, 1879, 523.