CHAP, i] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 35 earning aliquot parts, apart from the testing of the primality of a number n, knowing no method except the trial of each number < \/n as a divisor. Descartes312 gave the following rules for multiply perfect numbers: I. If n is a P3 not divisible by 3, then 3n is a P4. II. If a P3 is divisible by 3, but by neither 5 nor 9, then 45P3 is a P4, III. If a P3 is divisible by 3, but not by 7, 9 or 13, then 3-7-13 P3 is a P4. IV. If n is divisible by 29, but by no one of the numbers 210, 31, 43, 127, then 31n and 1643-127n are proportional to the sums of their aliquot parts. V. If n is not divisible by 3 and if 3n is a P4fc, then n is a P3ft. By applying rule II to P3(2), P3(3), P3(4), Descartes obtained his P4(1), P4(3), P4(5). By applying rule III to P3(1), P3(3), P3(4), he obtained his P4<2), P4«, P4(6>. In the same letter, Descartes expressed to Mersenne a desire to know what Frenicle de Bessy had found on this subject. Frenicle wrote direct to Descartes, who in his reply313 expressed his astonishment that Frenicle should regard as sterile the above rules for finding P4, since Descartes had deduced by them six P4 from four P3, at a time when Mersenne had stated to Descartes that it was thought to be impossible to find any at all. Descartes stated that, since one can find an infinity of such rules, one has the means of finding an infinitude of Pm. From one of Frenicle's P$ (communicated to Descartes by Mersenne), P5<2) = 30823866178560=210355-7213-19-23-89, Descartes (p. 475) derived the smaller P5: P5(3) = 31998395520 = 27355-72-13-17-19. Mersenne314 listed various Pm due to his correspondents, without citation of names. He listed the above P3(t) (i = 1, 2, 3, 4) and remarked that "un excellent esprit"315 found that when P3(5) =459818240=285-7-19-37-73 is multiplied by 3, the product is a P4: P4(7) = 283.5-7.19-37-73, attributed to Lucas321 by Carmichael.334 8120euvres, 2, 1898, 427-9, letter to Mersenne, Nov. 15, 1638. '"Oeuvres de Descartes, 2, 1898, 471, letter to Frenicle, Jan. 9, 1639. 814Les Nouvelles Pensees de Galilei, traduit d'ltalien en Francois, Paris, 1639, Preface, pp. 6-7. Quoted in Oeuvres de Descartes, 10, Paris, 1908, pp. 564-6, and in Oeuvres de Fermat, 4, 1912, pp. 65-66. mFrenicle de Bessy, according to the editors of the Oeuvres de Fermat, 2, 1894, p. 255, note 2; 4, 1912, p. 65, note 2 (citing Oeuvres de Descartes, 2, letter Descartes to Mersenne, Nov. 15, 1638, pp. 419-448 [p. 429]). It is clear that the discoverers Fermat, St. Croix, and Descartes of the P»(<) (i=2, 3,4) are not meant. It is attributed to Legendre11* by Carmichael.m