54 History of the Theory of Numbers. [ChapII 
a/S^ — 1; for A = 14, take B = 2, whence P = 5. Again, the sum of the 
aUquot parts of 3P~ is (2+-P)^. The numbers AP and BPQ have the same 
sum of divisors if a(P+l) = 6(P+1)(Q+1), i. e., if Q = a/6-l; takmg 
a = 24, 6 = 6, we have Q = 3, a prime, .4 = 14, B = 5 (by his table of the sum 
of the divisors of 1,. . ., 150); this problem had been solved otherwise by 
Wolff." 
L. Euler^^ gave a table of the prime factors of o-(p), a(p^), and <t(p^) for 
each prime p<1000; also those of aip") for various a's for p^23 (for 
instance, a ^36 when p = 2). He proved formulas (1) and (2) here and in 
his^ posthumous tract, where he noted (p. 514) all the cases in which 
a{n) =a-(?70 = 60. 
E. Waring^2 proved formula (2). He^ noted that if P = arir. . .and 
Q = a'^h^ . . . , where m — a,n — ^,... are large, then a{PQ)/a{P) is just greater 
than Q. If A = {1-1)\, <t{IA)/(t{A)^1-^1. If a'br..=A and (x+l) 
(2/+ 1) . . . is a maximum, then a'"''"^ = 6""^^ = . . . For a, 6, . . . distinct primes, 
a{A) is not a maximum. He cited numbers with equal sums of divisors: 
6 and 11, 10 and 17, 14 and 15 and 23. 
L. Kronecker^^ derived the formulas for the number and sum of the 
divisors of an integer by use of infinite series and products. 
E. B. Escott^^ listed integers whose sum of divisors is a square. 
Problems of Fer\la.t and Wallis on Sums of Divisors. 
Fermat'*^ proposed January 3, 1657, the two problems: (i) Find a cube 
which when increased by the sum of its aUquot parts becomes a square;* 
for example, 7^ + ( 1 + 7 + 7^) = 20^. (ii) Find a square which when increased 
by the sum of its aliquot parts becomes a cube. 
John WalUs^^ replied that unity is a solution of both problems and pro- 
posed the new problem: (m) Find two squares, other than 16 and 25, such 
that if each is increased by the sum of its ahquot parts the resulting sums 
are equal. 
Brouncker*^ gave 1/n^ and 343/n^ as solutions (!) of problem (i). 
"Elementa Analyseos, Cap. 2, prob. 87. 
«Opuscula varii argumenti, 2, Berlin, 1750, p. 23; Comm. Arith., 1, 102 (p. 147 for table to 100). 
Opera postuma, I, 1862, 95-100. F. Rudio, Bibl. Math., (3), 14, 1915, 351, stated that 
there are fully 15 errors. 
"Comm. Arith., 2, 512, 629. Opera postuma, I, 12-13. 
«Meditationea Algebr., ed. 3, 1782, 343. (Not in ed. of 1770.) 
"Vorlesungen iiber Zahlentheorie, I, 1901, 265-6. 
^Amer. Math. Monthly, 23, 1916, 394. 
*Erroneou8ly given as "cube" in the French tr., Oeuvres de Fermat, 3, 311. 
'"OeuvTes, 2, 332, "premier d6fi aux mathdmaticiens;" also, pp. 341-2, Fermat to Digby, June 
6, 1657, where 7' is said to be not the only solution. These two problems by Fermat 
were quoted in a letter by the Astronomer Jean H6v4hus, Nov. 1, 1657, pubhshed by C. 
Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 683-5, along with extracts from the 
Commercium EpistoUcum. Cf. G. Wertheim, Abh. Geschichte Math., 9, 1899, 558-561, 
570-2 ( = Zeitschr. Math. Phys., 44, Suppl. 14). 
"Commercium Epistohcum de Wallis, Oxford, 1658; Walhs, Opera, 2, 1693. Letter II, from 
WaUis to Brouncker, Mar. 17, 1657; letter XVI, Walhs to Digby, Dec. 1, 1657. Oeuvres 
de Fermat, 3, 404, 414, 427, 482-3, 503-4, 513-5. 
**Commercium, letter IX, Wallis to Digby; Fermat'e Oeuvres, 3, 419. 
