54 HlSTOKT OF THE THEORY OP NUMBERS. a/J52-l; for A = 14, take 5 = 2, whence P=5. Again, the sum of the aliquot parts of 3P2 is (2+P)2. The numbers AP and BPQ have the same sum of divisors if a(P+l) = 6(P+l)(Q+l), i. 6., if Q = a/6-l; taking a = 24, b=6, we have Q = 3, a prime, A = 14, 5=5 (by his table of the sum of the divisors of 1,..., 150); this problem had been solved otherwise by Wolff.31 L. Euler32 gave a table of the prune factors of <r(p), <r(;p2), and <7(p3) for each prime p<1000; also those of <r(p°) for various a's for p^23 (for instance, a^36 when p=2). He proved formulas (1) and (2) here and in his33 posthumous tract, where he noted (p. 514) all the cases in which <r(n)=<r()tt)^60. E. Waring12 proved formula (2). He34 noted that if P=om6n.. .and Q =aa6*..., where w-a, n~/3,.. .are large, then (r(PQ)/<r(P) is just greater than Q. If A = (Z-1)!, <r(Li)/<r(A)gZ+l. If oV...=4 and (s+1) (l/4-1).. .is a maximum, then az+1=6V+1 = ... For a, 6,... distinct primes, or (-4.) is not a maximum. He cited numbers with equal sums of divisors: 6 and 11,10 and 17, 14 and 15 and 23. L. Kronecker35 derived the formulas for the number and sum of the divisors of an integer by use of infinite series and products. E. B. Escott36 listed integers whose sum of divisors is a square. PROBLEMS OF FERMAT AND WALLIS ON SUMS OF DIVISORS. Fermat40 proposed January 3, 1657, the two problems: (i) Find a cube which when increased by the sum of its aliquot parts becomes a square;* for example, 73 + (1+7+72) = 202. (w) Find a square which when increased by the sum of its aliquot parts becomes a cube. John Wallis41 replied that unity is a solution of both problems and proposed the new problem: (Hi) Find two squares, other than 16 and 25, such that if each is increased by the sum of its aliquot parts the resulting sums are equal. Brouncker42 gave 1/n6 and 343/ne as solutions (!) of problem (i). "Elements Analyseos, Cap. 2, prob. 87. ^pusculavarii argument*, 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. wComm. Arith., 2, 512, 629. Opera postuma, I, 12-13. **Meditationes Algebr., ed. 3, 1782, 343. (Not in ed. of 1770.) "Vorlesungen iiber Zahlentheorie, I, 1901, 265-6. ťAmer. Math. Monthly, 23, 1916, 394. 'Erroneously given as "cube" in the French tr., Oeuvres de Fermat, 3, 311. 400euvres, 2,332, "premier de"fi aux mathematiciens;" 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 Heve*lius, Nov. 1, 1657, published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fis., 12, 1879, 683-5, along with extracts from the Commercium Epistolicum. Cf. G. Wertheim, Abh. Geschichte Math., 9, 1899, 558-561, 570-2 (=Zeitschr. Math. Phys., 44, Suppl. 14). K^ommercium Epistolicum de Wallis, Oxford, 1658; Wallis, Opera, 2, 1693. Letter II, from Wallis to Brouncker, Mar. 17,1657; letter XVI, Wallis to Digby, Dec. 1. 1657. Oeuvres de Fermat, 3, 404, 414, 427, 482-3, 503-4, 513-5. ^ommercium, letter IX, Wallis to Digby; Fennat'fl Oeuvres, 3, 419.