34 History of the Theory of Numbers. [Chap. I 
progression 2, 4, 8, ... . Subtract unity and place the remainders above the 
former. Add unity and place the sums below. Then if the quotient of 
the (n+3)th number of the top line by the nth number of the bottom 
line is a prime, its triple multiplied by the (n+2)th number of the middle 
line is a P3. Thus if ?i = l, 15/3 is a prune and 3-5-8 = 120 is a P3. For 
n = 3, 63/9 is a prime and 3-7-32 = 672 is a P3. [This rule thus states in 
effect that 3-2"+2p is a P3 if p = (2'»+3-l)/(2"+l) is a prune.] 
The third P3, discovered by Andr6 Jumeau, Prior of Sainte-Croix, is 
P3^'^ = 523776 = 29311-31. 
In April, 1638, he communicated it to Descartes^°^ and asked for the fourth 
P3 (the fifth and last of St. Croix's challenge problems). 
Descartes^*^^ stated that the rule ^^^ of Fermat furnishes no P3 other than 
120 and 672 and judged that Fermat did not find these numbers by the 
formula, but accommodated the formula to them, after finding them by trial. 
Descartes^^^ answered the challenge of St. Croix with the fourth P3, 
P3^'^ = 1476304896 = 2^33.1143.127. 
Soon afterwards Descartes^ °^ announced the following six P4: 
P4^i) = 30240 = 2^335.7, 
P4(2) =32760 = 2^325.7.13, 
P^^^^ = 23569920 = 2^335. 1 1 -31 , 
P4(*) = 142990848 = 2^327.11.13.31, 
P4(^> = 66433720320 = 2^^335. 1 1 .43. 1 27, 
P4^®^ =403031236608 = 2^3327.11.13.43.127, 
and the sous-quadruple 
Ps^^^ = 14182439040 = 2^3^5.7-11217.19. 
He stated that his analysis had led him to a method which would require 
time to explain in the form of a rule, but that he could find, for example, 
a sous-centuple, necessarily very large. 
Fermat apparently responded to the fifth challenge problem of St. Croix 
on the fourth P3. Without warrant, Descartes^^" suspected that Fermat 
had not found independently the fourth P3, but had learned from some one 
in Paris of its earlier discovery by Descartes. Fermat^^^ indicated that he 
possessed an analytic method by which he could solve all questions con- 
'"^Oeuvres de Descartes, 2, Paris, 1898, p. 428, p. 167 (latter without name of St. Croix); cf. 
Oeuvres de Fermat, 2, 1894, pp. 63-64. 
"'Oeuvres de Descartes, 2, 1898, p. 148, letter to Mersenne, May 27, 1638. 
^osQeuvres de Descartes, 2, 1898, 167, letter to Mersenne, June 3, 1638. 
'"'Oeuvres de Descartes, 2, 1898, 2.50-1, letter to Mersenne, July 13, 1638. In June, 1645, 
Descartes, 4, 1901, p. 229, again mentioned the first two of these Pt. 
""Oeuvres de Descartes, 2, 1898, 273, letter to Mersenne, July 27, 1638. 
"^Oeuvres de Fermat, 2, 1894, p. 165, No. 4; p. 176, No. 1; letters to Mersenne, Aug. 10 and 
Dec. 26, 1638. 
