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36 HISTORY OF THE THEORY OF NUMBERS. [CHAP, i There are listed Descartes' six P4 and P5(1), Frenicle's P5(2), and also P4<8) = 45532800=27335217-31, P4(9) = 43861478400 = 210335223-31-89, and the erroneous P5 508666803200 (not divisible by 52+5+l), probably a misprint for the correct P5 (in the list by Lehmer323): P5{4) = 518666803200 = 2U33527213-19-31. A part of these PTO, but no new ones, were mentioned by Mersenne60 in 1644; the least P3 is stated to be 120. (Oeuvres de Fermat, 4, 66-7.) In 1643 Fermat316 cited a few of the Pm he had found: P3(6) = 51001180160=2145-749.3M51, P4(10)-3P3(6), P4(n) = 14942123276641920 = 27365-17-23-137.547-1093, P5<5> = 1802582780370364661760=220335.7213219-31.61.127.337, P5w = 87934476737668055040 = 217355-7313-19237.73.127, P6(1) = 2233W1131331723141.61.241.307467.2801, P6(2)=2273553741.13219.29.3143-61.113-127. He stated that he possessed a general method of finding all PTO. Replying to Mersenne's query as to the ratio of P6(3)=23638561143219.31243.61.83.223.331.379.601.757 X 1201.7019-823543-616318177400895598169 to the sum of its aliquot parts, Fermat317 stated that it is a P6, the prime factors of the final factor being 112303 and 898423 [on the finding of these factors, see Ch. XIV, references 23, 92, 94, 103]. Note that 823543 = 77. Descartes318 constructed P3(2)=672=21.32 by starting with 21 and noting that <r(21) =32, (r(32) = 63 = 3-21, for <r defined as on p. 53. Mersenne61 noted that if a P3 is not divisible by 3, then 3P3 is a P4 [rule I of Descartes312]; if a P5 is not divisible by 5, then 5P5 is a P6, etc. He stated that there had been found 34 P4,18 P5,10 P6,7 P7,but no P8so far. In 1652, J. Broscius (Apologia,64 p. 162) cited the P4(1) [of Descartes309]. The P3 120 and 672 are mentioned in the 1770 edition of Ozanam's79 Recreations, I, p. 35, and in Hutton's translation of Montucla's" edition, I, p. 39. A. M. Legendre319 determined the Pm of the form 2na.fty..., where a, /3, 7,.. .are distinct odd primes, for ?w = 3, n^8; w = 4, n=3, 5; w = 5, w = 7. No new Pn were found. 3160euvres, 2,1894, p. 247 (261), letter to Carcavi; Varia opera, p. 178; Precis des oeuvrea math, de Fermat, par E. Brassinne, Toulouse, 1853, p. 150. 3UOeuvers de Fermat,-2,1894,255, letter to Mersenne, April 7,1643. The editors (p. 256, note) explained the method of factoring probably used by Fermat. The sum of the aliquot parts of 236 is 223.Y, where N = 616318177, and the sum of the aliquot parts of N is 2-7? M M=898423. As M does not occur elsewhere in P«, it is to be expected as a factor of the final factor of P«. '"Manuscript published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12,1879. 714. "Thor&e desnombres, 3d ed., vol. 2, Paris, 1830, 146-7; German transl. by H. Maser, Leipzig, 2, 1893, 141-3. The work for m=3 was reproduced by Lucas320 without reference.