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 n-1, 15/3 is a prime 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 3-2n+2p is a P3 if p=(2n+3-l)/(2n+l) is a prime.] The third P3, discovered by Andr6 Jumeau, Prior of Sainte-Croix, is P3(3) = 523776 = 29341-31. In April, 1638, he communicated it to Descartes306 and asked for the fourth P3 (the fifth and last of St. Croix' s challenge problems). Descartes307 stated that the rule 306 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. Descartes308 answered the challenge of St. Croix with the fourth P3, P3(4) = 1476304896 = 21334143-127. Soon afterwards Descartes309 announced the following six P4: P4(1) = 30240 = 25335-7, P4(2)= 32760 = 23325-7- 13, P4(3) = 23569920 = 29335-ll-31, P4(4) = 142990848 = 29327-lM3-31, P4(5) = 66433720320 = 213335-1 1-43- 127, P4(6)=403031236608 = 213327.11.1343427, and the sous-quadruple Ps(1) = 14182439040 = 27345-7.11217-19. He stated that his analysis had led him to a method which would require time to explain hi 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, Descartes310 suspected that Fermat had not found independently the fourth P3, but had learned from some one in Paris of its earlier discovery by Descartes. Fermat311 indicated that he possessed an analytic method by which he could solve all questions con- 8MOeuvres de Descartes, 2, Paris, 1898, p. 428, p. 167 (latter without name of St. Croix); cf. Oeuvresde Fermat, 2, 1894, pp. 63-64. «07Oeuvres de Descartes, 2, 1898, p. 148, letter to Mersenne, May 27, 1638. «08Oeuvres de Descartes, 2, 1898, 167, letter to Mersenne, June 3, 1638. 80BOeuvres de Descartes, 2, 1898, 250-1, letter to Mersenne, July 13, 1638. In June, 1645, Descartes, 4, 1901, p. 229, again mentioned the first two of these P4. 8100euvres 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.