52 HlSTOBY OF THE THEORY OF NUMBERS. [CHAP. II factors are r, s,..., the required number is pr~lq*~l..., where p, g,.. .are any distinct primes. When the number of divisors is odd, the number itself is a square, and conversely. The number of waysN=a*bft.. .canbe expressed as a product of two factors is fc=^(a+l)(/3+l).. .or $+k, according as N is not or is a square. Jean Prestet9 noted that a product of k distinct primes has 2* divisors, while the nth power of a prime has n-fl divisors. The divisors of a263c2 are the 12 divisors of a263, their products by c and by c2, the general rule not being stated explicitly. Pierre R&nond de Montmort10 stated in words that the number of divisors of aa€l.. .a/" is fo+1).. .(en+l) if the a's are distinct primes. Abbe* Deidier11 noted that a product of k distinct primes has divisors, treating the problem as one on combinations (but did not sum the series and find 2*). To find the number of divisors of 243352 he noted that five are powers of 2 (including unity). Since there are three divisors of 33, multiply 5 by 3 and add 5, obtaining 20. In view of the two divisors of 52, multiply 20 by 2 and add 20. The answer is 60. E. Waring12 proved that the number of divisors of ambn.. .is (w+1) (n+1).. .if a, 6,.. .are distinct primes, and that the number is a square if the number of its divisors is odd. E. Lionnet13 proved that if a, 6, c,.. .are relatively prime in pairs, the number of divisors of abc.. .equals the product of the number of diyisors of a by the number for b, etc. According as a number is a square oij not, the number of its divisors is odd or even. T. L. Pujo14 noted the property last mentioned. Emil Hain16 derived the last theorem from am = (^... tm)2, where £1,..., tm denote the divisors of a. A. P. Minin16 determined the smallest integer with a given number of divisors. G. Fontene*17 noted that, if 2a3*. . .mMn" (a^/3^ ... ^^v) is the least number with a given number of divisors, then v+1 is a prime, and ju+1 is a prune except for the least number 233 having eight divisors. FORMULA FOE THE SUM OF THE DIVISORS OF A NUMBER. R. Descartes,21 in a manuscript, doubtless of date 1638, noted that, if p is a prime, the sum of the aliquot parts of pn is (pn— l)/(p — 1). If b is the »Nouv. Elemens des Math., Paris, 1689, vol. 1, p. 149. "Essay d'analyse sur lea jeux de hazard, ed. 2, Paris, 1713, p. 55. Not in ed. 1, 1708. "Suite de 1'arithme'tique des ge'ome'tres, Paris, 1739, p. 311. "Medit. Algebr., 1770, 200; ed. 3, 1782, 341. "Nouv. Ann. Math., (2), 7, 1868, 68-72. 14Les Mondes, 27, 1872, 653-4. "Archiv Math. Phys., 55, 1873, 290-3. "Math. Soc. Moscow (in Russian), 11, 1883-4, 632. "Nouv. Ann. Math., (4), 2, 1902, 288; proof by Chalde, 3, 1903, 471-3. llt(De partibus aliquotis mimerorum," Opuscula Posthuma Phye. et Math., Amstelodami, 1701, p. 5; Oeuvres de Descartes (ed. Tannery and Adams, 1897-1909), vol. 10, pp. 300-2.