CHAP, ii] PROBLEMS ON SUMS OF DIVISORS. 56 Frenicle43 expressed his astonishment that experienced mathematicians should not hesitate to present, for the third tune, unity as a solution. Wallis44 tabulated <r(z?) for each prime #<100 and for low powers of 2,3,5, and then excluded those prunes sfor which cr(x3)has a prime factor not occurring elsewhere in the table. By similar eliminations and successive trials, he was led to the solutions48 of (i): a=3354M34147, 6=2-3-5-134147; 7a, 76, adding that they are identical with the four numbers given by Frenicle.46 Note that <r(a) is the square of 28325-7-lM3-17-29-61, while cr(b) is the square of 2732527-13-17-29. Wallis47 gave the further solutions of o-(x3)=y2: x= 17-3147491, 2/=210325-13-17-29-37, 2-3-5-13-17-314M91, 21233527-13-17-29*37, 335-ll-13-17-314M91, 213335-7-lM3-17-29237-61, and the products of each x by 7. Wallis48 gave solutions of his problem (Hi): 233-37, 2-19-29; 223-ll-19-37, 237-29-67; 29-67, 2-3-5-37; 237-29-67, 3-5-11-19-37. Frenicle49 gave 48 solutions of Wallis' problem (Hi), including 2-163 11-37; 3-11-19, 7-107; 2-5-151, 33-67; also 83 sets of three squares having the same sum of divisors, for example, the squares of 22ll-37-151, 3367-163, 5-11-37-151, o-=327319-31-67-1093; also various such sets of n squares (with prune factors <500) for n^!9, for example, the squares of ac, ad, 4bd, 46c, 56d, and 56c, where a = 2.5-2947-67-139, 6 = 13-37-191-359, c=7-107, d = 3-lM9. Frans van Schooten60 made ineffective attempts to solve problems (0, (ii). Frenicle51 gave the solution z = 225.7-11.37-67-163-191-263-439499, 2/ = 327313-19-31267.109 of problem (ii), <r(x2) =?/3; also a new solution of <r(x3) = y2: x = 255-7-31 -73-241 -243-467, y = 212325311 -13217-37-41 • 113-193-257. "Letter XXII, to Digby, Feb. 3, 1658. Cf. Leibnitii et Bernoulli! Commercium philos. et math., I, 1795, 263, letter from Johann Bernoulli to Leibniz, Apr. 3, 1697. "Letter XXIII, to Digby, Mar. 14, 1658. aThe same tentative process for finding this solution a was given by E.Waring, Meditationes Algebraicae, 1770, pp. 216-7; ed. 3, 1782, 377-8. The solution 6=751530 was quoted by Lucas, The"orie des nombres, 1891, 380, ex. 3. 48Solutio duonim problematum circa numeros cubos.. .1657, dedicated to Digby [lost work]. See Oeuvres de Format, p. 2. 434, Note; Wallis.68' "Letter XXVIII, March 25, 1658; Wailis, Opera, 2, 814; Wallis". "Letter XXIX, Mar. 29, 1658; Wallis". "Letter XXXI, Apr. 11, 1658. "Letter XXXIII, Feb. 17, 1657 and Mar. 18, 1658. "Letter XLIII, May 2, 1658.