36 History of the Theory of Numbers. [Chap, i 
There are listed Descartes' six P^ and P^^^^ Frenicle's Ps^\ and also 
P4^«) =45532800 = 2^3^5217.31, 
P4^^^ = 43861478400 = 2^°3^5223.31.89, 
and the erroneous P5 508666803200 (not divisible by 5^+5-|-l), probably 
a misprint for the correct P5 (in the list by Lehmer^'^) : 
P5^'^ = 518666803200 = 2^^3'5-72l3- 19-31 . 
A part of these Pm, but no new ones, were mentioned by Mersenne^" in 
1644; the least P3 is stated to be 120. (Oeuvres de Fermat, 4, 66-7.) 
In 1643 Fermat^ ^^ cited a few of the P^ he had found: 
Ps^^^ = 51001 180160 = 2^*5.7.19.31.151 , 
P4^'^) = 14942123276641920 = 2^3^5.17.23.137.547.1093, 
P5(^) = 1802582780370364661760 = 22°335.72l32l9.31.61. 127.337, 
Ps^®^ = 87934476737668055040 = 2^^3^5.7313. 19-37.73. 127, 
Pg(i) = 223375374113133^7231.41.^1.241.307.467.2801, 
Pg(2) = 22735537.11.13219.29.31.43.61.113.127. 
He stated that he possessed a general method of finding all P^n- 
Replying to Mersenne's query as to the ratio of 
Pg(3) = 22^3^5^11.13219.31243.61.83.223.331.379.601.757 
X 1201.7019.823543-616318177:100895598169 
to the sum of its aUquot parts, Fermat^^^ stated that it is a Pq, the prime 
factors of the final factor being 112303 and 898423 [on the finding of these 
factors, see Ch. jXIV, references 23, 92, 94, 103]. Note that 823543 = 7^ 
Descartes^^^ constructed P3^2) = 572 = 21.32 by starting with 21 and 
noting that (r(21) =32, o-(32) =63 = 3.21, for a defined as on p. 53. 
Mersenne^^ noted that if a P3 is not divisible by 3, then 3P3 is a P4 
[rule I of Descartes^^-]; if a P5 is not divisible by 5, then 5P5 is a Pq, etc. 
He stated that there had been found 34 P4, 18 P5, 10 Pg, 7 P7, but no Pgso far. 
In 1652, J. Broscius (Apologia,^* p. 162) cited the P4^'^ [of Descartes^"^]. 
The P3 120 and 672 are mentioned in the 1770 edition of Ozanam's'^ 
Recreations, I, p. 35, and in Hutton's translation of Montucla's^^ edition, 
I, p. 39. 
A. M. Legendre^^^ determined the Pm of the form 2"a/37 . . . , where a, 
/3, 7, . . .are distinct odd primes, for 7n = 3, n^8; ?n=4, n = 3, 5; m = 5, n = 7. 
No new P„ were found. 
"*Oeuvres, 2, 1894, p. 247 (261), letter to Carcavi; Varia opera, p. 178; Pr^cia des oeuvres math, 
de Fermat, par E. Brassinne, Toulouse, 1853, p. 150. 
3'^Oeuvers 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 23« is 223iV, where N = 616318177, and the sum of the aliquot parts of N is 2-7? M 
iVf = 898423. As M does not occur elsewhere in Pe., it is to be expected as a factor of 
the final factor of Pe. 
"8Manuscript published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 714. 
'^•Thor^ie des nombres, 3d ed., vol. 2, Paris, 1830, 146-7; German transl. by H. Maser, Leipzig, 
2, 1893, 141-3. The work for n? =3 was reproduced by Lucas'^" without reference. 
