52 History of the Theory of Numbers. [Chap, ii 
factors are r, s, . . ., the required number is p''~^q'~^ . . ., where p, q,. . are 
any distinct primes. WTien the number of divisors is odd, the number 
itself is a square, and conversely. The number of ways A^ = a^b^ . . . can be 
expressed as a product of two factors is A = |(a+l)(j3+l) . . .or \-\-k, 
according as N is not or is a square. 
Jean Prestet^ noted that a product of k distinct primes has 2'' divisors, 
while the ?ith power of a prime has n+1 divisors. The divisors of a^h^c^ 
are the 12 divisors of or}?, their products by c and by c^, the general rule 
not being stated explicitly. 
Pierre R^mond de Montmort^° stated in words that the number of 
divisors of Oi**. . .a„*" is (ci+l) . . .(e„+l) if the a's are distinct primes. 
Abb^ Deidier^^ noted that a product of k distinct primes has 
^+^+(2) + (3)+ • 
divisors, treating the problem as one on combinations (but did not sum the 
series and find 2*"). To find the number of divisors of 2*3^5^ he noted that 
five are powers of 2 (including unity). Since there are three divisors of 3^, 
multiply 5 by 3 and add 5, obtaining 20. In view of the two divisors of 
5^, multiply 20 by 2 and add 20. The answer is 60. 
E. Waring^- proved that the number of divisors of a"'?)". . .is (m+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. Lionnet^^ proved that if a, b, c, . . .are relatively prime in pairs, the 
number of divisors of abc. . .equals the product of the number of divisors 
of a by the number for b, etc. According as a number is a square or not, 
the number of its divisors is odd or even. 
T. L. Pujo^^ noted the property last mentioned. 
Emil Hain^^ derived the last theorem from a"* = (<i . . . t„y, where <i, . . . , <„ 
denote the divisors of a. 
A. P. Minin^^ determined the smallest integer with a given number of 
divisors. 
G. Fontene'" noted that, if 2"3^. . .mV (a^/S^ . . . ^n^v) is the least 
number with a given number of di\4sors, then I'+l is a prime, and /x+1 is 
a prime except for the least number 2^3 ha\'ing eight di\'isors. 
Formula for the Sum of the Divisors of a Number. ' 
R. Descartes,^^ in a manuscript, doubtless of date 1638, noted that, if p 
is a prim6, the sum of the aliquot parts of p" is (p"— l)/(p — 1). If 6 is the 
"Nouv. Elemens des Math., Paris, 1689, vol. 1, p. 149. 
loEssay d'analyse sur les jeux de hazard, ed. 2, Paris, 1713, p. 55. Not in ed. 1, 1708. 
"Suite de I'arithm^tique des g^om^tres, Paris, 1739, p. 311. 
i^Medit. Algebr., 1770, 200; ed. 3, 1782, 341. 
"Nouv. Ann. Math., (2), 7, 1868, 68-72. 
"Les 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. 
*'"De partibus ahquotis numerorum," Opuscula Posthuma Phys. et Math., Amstelodami, 
1701, p. 5; Oeuvres de Descartes (ed. Tannery and Adams, 1897-1909), vol. 10, pp. 300-2. 
