CHAPTER II. 
FORMULAS FOR THE NUMBER AND SUM OF DIVISORS. PROBLEMS OF 
FERMAT AND WALLIS. 
Formula for the Number of the Divisors of a Number. 
Cardan^ stated that a product P oi k distinct primes has 1+2+2^+ . . 
-\-2^~'^ aUquot parts (divisors <P). 
Michael StifeP proved this rule and found^ the number of divisors of 
2*3^52p, where P = 7-11.13-17-19-23-29, by first noting that there are 
1+2+ . . .+64 divisors <P oi P according to Cardan's rule and hence 
128 divisors of P. The factor 5^ gives rise to 128+128 more divisors, so 
that we now have 384 divisors. The factor 3^ gives 3.384 more, so that we 
have 1536. Then the factor 2* gives 4.1536 more. 
Mersenne* asked what number has 60 divisors; since 60 = 2-2-3-5, sub- 
tract unity from each prime factor and use the remainders 1, 1, 2, 4 as 
exponents; thus 3^-2*-7-5 = 5040 (so much lauded by Plato) has 60 divisors. 
It is no more difficult if a large number of aliquot parts is desired. 
I. Newton^ found all the divisors of 60 by dividing it by 2, the quotient 
30 by 2, and the new quotient 15 by 3. Thus the prime divisors are 1, 2, 2, 
3, 5. Their products by twos give 4, 6, 10, 15. The products by threes 
give 12, 20, 30. The product of all is 60. The commentator J. Castillionei, 
of the 1761 edition, noted that the process proves that the number of all 
divisors of a'"6'*. . .is (m+l)(n+l) . . .if a, 6, . . .are distinct primes. 
Frans van Schooten^ devoted pp. 373-6 to proving that a product of k 
distinct primes has 2'''— 1 aliquot parts and made a long problem (p. 379) 
of that to find the number of divisors of a given number. To find (pp. 
380-4) the numbers having 15 aliquot parts, he factored 15+1 in all ways 
and subtracted unity from each factor, obtaining abed, a^bc, a%^, a^b, a^^. 
By comparing the arithmetically least numbers of these various types, he 
found (pp. 387-9) the least number having 15 aliquot parts. 
John Kersey'^ cited the long rule of van Schooten to find the number of 
aliquot parts of a number and then gave the simple rule that Oi" . . . a^" has 
(6i+l) . . . (e„+l) divisors in all if ai, . . . , a„ are distinct primes. 
John Wallis^ gave the last rule. To find a number with a prescribed 
number of divisors, factor the latter number in all possible ways; if the 
iPractica Arith. & Mensurandi, Milan, 1537; Opera, IV, 1663. 
*Arithmetica Integra, Norimbergae, 1544, lib. 1, fol. 101. 
'Stifel's posthumous manuscript, fol. 12, preceding the printed text of Arith. Integra; cf. E. 
Hoppe, Mitt. Math. Gesell. Hamburg, 3, 1900, 413. 
*Cogitata Physico Math., II, Hydravhca Pnevmatica, Preface, No. 14, Paris, 1644. (Quoted 
by Winsheim, Novi Comm. Ac. Petrop., II, ad annum 1749, Mem., 68-99). Also letter 
from Mersenne to Torricello, June 24, 1644, Bull. Bibl. Storia Sc. Mat., 8, 1875, 414-5. 
•Arithmetica UniversaUs, ed. 1732, p. 37; ed. 1761, I, p. 61. De Inventione Divisorum. 
•Exercitationum Math., Lugd. Batav., 1657. 
^The Elements of Algebra, London, vol. 1, 1673, p. 199. 
•A Treatise of Algebra, London, 1685, additional treatise, Ch. III. 
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