Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 35 
cerning aliquot parts, apart from the testing of the primaUty of a number n, 
knowing no method except the trial of each number < \/n as a divisor. 
Descartes"^ 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 2^ but by no one of the numbers 2^°, 31, 43, 127, 
then 31n and 16-43-127n are proportional to the sums of their 
aliquot parts. 
V. If n is not divisible by 3 and if 3n is a P^k, then n is a Ps^. 
By applying rule II to P3^^\ Ps^^\ P3^*^ Descartes obtained his P^^^^ 
P4<3^ P^^^K By applying rule III to Ps^'\ Ps^^\ Ps^''\ he obtained his 
p (2) p (4) p (6) 
^4 ) ^4 5 ^4 • 
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 reply^^^ 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. Des- 
cartes stated that, since one can find an infinity of such rules, one has the 
means of finding an infinitude of P^- From one of Frenicle's P5 (com- 
municated to Descartes by Mersenne), 
Ps^^) = 30823866178560 = 2i°3^5-72l3- 19-23-89, 
Descartes (p. 475) derived the smaller P5: 
Pg^^) = 31998395520 = 2^3^5.72.13.17.19. 
Mersenne^^^ listed various P^ due to his correspondents, without cita- 
tion of names. He listed the above Ps^'^ (^ = 1, 2, 3, 4) and remarked that 
"un excellent esprit "^^^ found that when 
P3^'^ = 459818240 = 2^5.7.19.37.73 
is multiplied by 3, the product is a P4: 
P4(^ = 2^3.5.7.19.37.73, 
attributed to Lucas^^^ by Carmichael.^^^ 
"^Oeuvres, 2, 1898, 427-9, letter to Mersenne, Nov. 15, 1638. 
"'Oeuvres de Descartes, 2, 1898, 471, letter to Frenicle, Jan. 9, 1639. 
'"Les Nouvelles Pensees de Galilei, traduit d'ltalien en Frangois, 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. 
'"Frenicle 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 Legendre"" by 
Carmichael.*'* 
