23 
3 
13 
Chap. II] FORMULAS FOR NUMBER AND SUM OF DiVISORS. 53 
sum of the aliquot parts of a, the sum of the ahquot parts of ap is 6p+a+6. 
If b is the sum of the ahquot parts of a and if x is prime to a, the sum of the 
aUquot parts of ax"" is 
— ^n — [=^'+^K-j^)-^^\' 
Descartes^^ stated a result which may be expressed by the formula 
(1) <j{nm)=(T{n)(T{m) (n, m relatively prime), 
where (T{n) is the sum of the divisors (including 1 and n) of w. Here he 
solved n : (T(n) = 5 : 13. Thus n must be divisible by 5. Enter 5 
in column A and (r(5) = 6 in column B. Then enter the factor A B 
2 in column A and (r(2) =3 in column B. Having two threes 
in column B, we enter 9 in column A and cr(9) = 13 in B. Every 5 
number except 13 in column B is in column A. Hence the 2 
product 5-2-9 = 90 is a solution n. Next, to solve n : a- (n) = 5 : 14, 9 
we enter also 13 in column A and 14 in B, and obtain the solu- 
tion 90-13. If ?i is a perfect number, 5n: (7(5n) = 5: 12 and, if n?^6, 15n: 
(r(15n) = 5:16. 
Descartes^^ stated that he possessed a general rule [illustrated above] 
for finding numbers having any given ratio to the sum of their aliquot parts. 
Fermat^^ had treated the same problem. Replying to Mersenne's 
remark that the sum of the aliquot parts of 360 bears to 360 the ratio 9 to 4, 
Fermat^^ noted that 2016 has the same property. 
John Wallis^^ noted that Frenicle knew formula (1). 
Wallis^^ knew the formula 
(2) ^(a•6^...) = 211^1-*^^.... 
a— 1 0—1 
Thus these formulae were known before 1685, the date set by Peano,^^ who 
attributed them to Wallis.^^ 
G. W. Kraft^^ noted that the method of Newton^ shows that the sum of 
the divisors of a product of distinct primes P, . . ., S is (P+1) . . .(S+1). 
He gave formula (1) and also (2), a formula which Cantor^" stated had 
probably not earlier been in print. To find a number the sum of whose 
divisors is a square, Kraft took PA, where P is a prime not dividing A. 
If (r(A)=a, then (r(PA) = (P-f-l)a will be the square of (P+l)5 if P = 
^^"De la fagon de trouver le nombres de parties aliquotes in ratione data," manuscript Fonds- 
frangais, nouv. acquisitions, No. 3280, ff. 156-7, Bibliothfeque Nationale, Paris. Pub' 
Ushed by C. Henry, BuU. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 713-5. 
^Oeuvres, 2, p. 149, letter to Mersenne, May 27, 1638. 
K)euvre8 de Fermat, 2, top of p. 73, letter to Roberval, Sept. 22, 1636. 
"Oeuvres, 2, 179, letter to Mersenne, Feb. 20, 1639. 
''^ommercium Epistolicum, letter 32, April 13, 1658; French transl. in Oeuvres de Fermat, 3, 
553. 
"Commercium Epist., letter 23, March, 1658; Oeuvres de Fermat, 3, 515-7. 
"Formulaire Math., 3, Turin, 1901, 100-1. 
"Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, 100-109. 
"Geschichte Math., 3, 595; ed. 2, 616. 
