# Full text of "History Of The Theory Of Numbers - I"

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```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,