Chap. II] PkOBLEMS ON SUMS OF DiVISORS. 55 
Frenicle^^ expressed his astonishment that experienced mathematicians 
should not hesitate to present, for the third time, unity as a solution. 
Wains'*^ tabulated <r{x^) for each prime a:<100 and for low powers of 
2, 3, 5, and then excluded those primes a: for which (T(rc^)has a prime factor 
not occurring elsewhere in the table. By similar eliminations and successive 
trials, he was led to the solutions^^ of (i) : 
a=3^5-lM3-41-47, 6=2-3-5-13-41-47; 7a, 7b, 
adding that they are identical with the four numbers given by Frenicle.*'' 
Note that o-(a) is the square of 2^3^5-7-lM3-17-29-61, while a{b) is the square 
of 2^3V7-13-17-29. Wallis^^ gave the further solutions of a{x^) = y^: 
a:=17-3147-191, ?/=2^°325-13-17-29-37, 
2-3-5-13-17.314M91, 2^^33527. i3.;^7.29237^ 
3'5-lM3-17-31-4M91, 2'23^5-7-lM3-17-29237-61, 
and the products of each x by 7. 
Wallis^^ gave solutions of his problem (m) : 
2^3.37, 2-19-29; 223-1M9-37, 2'7-29.67; 
29-67, 2-3-5-37; 2^7.29.67, 3-5-1M9-37. 
Frenicle*^ gave 48 solutions of WalUs' problem (m), including 2-163 
11-37; 3-11-19, 7-107; 2-5-151, 3^-67; also 83 sets of three squares having the 
same sum of divisors, for example, the squares of 
2^11.37-151, 3^67-163, 5-11-37-151, (7 = 3^7^19-31-67-1093; 
also various such sets of n squares (with prime factors <500) for n^l9, 
for example, the squares of ac, ad, 4:bd, 46c, 5bd, and 56c, where 
a = 2-5-29-47-67-139, 6 = 13-37-191-359, c = 7.107, d = 3-ll-19. 
Frans van Schooten^" made ineffective attempts to solve problems 
(i), (n). 
Frenicle^^ gave the solution 
a: = 225-7.11-37-67-163-191-263-439-499, t/ = 327^3- 19-31^67- 109 
of problem (n), a{x^)=y^; also a new solution of a{x^)=y^: 
a; = 255-7-31-73-241-243-467, i/ = 2i23253ii. 13217.37.41. 113.193.257. 
^'Letter XXII, to Digby, Feb. 3, 1658. Cf. Leibnitii et BernouUii Commercium philos. et 
math., I, 1795, 263, letter from Johann Bernoulli to Leibniz, Apr. 3, 1697. 
"Letter XXIII, to Digby, Mar. 14, 1658. 
"The 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, Thiorie des nombres, 1891, 380, ex. 3. 
**Solutio duorum problematum circa numeros cubos . . . 1657, dedicated to Digby [lost work]. 
See Oeuvres de Fermat, p. 2. 434, Note; WaUis." 
"Letter XXVIII, March 25, 1658; WaUis, Opera, 2, 814; Wallia". 
««Letter XXIX, Mar. 29, 1658; WaUis^^. 
"Letter XXXI, Apr. 11, 1658. 
"Letter XXXIII, Feb. 17, 1657 and Mar. 18, 1658. 
"Letter XLIII, May 2, 1658. 
