Chap. II] PROBLEMS ON SUMS OF DiVISORS. 57 
E. Fauquembergue,^^ after remarking that (1) is equivalent to the sys- 
tem (2), cited Fermat's^" assertion that the first two equations (2) hold 
only for a: = 7 [aside from the evident solutions a: = ± 1, 0], which has been 
proved by Genocchi.^^ 
H. Brocard^^ thought that Fermat's assertion that 7^ is not the only 
solution of problem (i) implied a contradiction with Genocchi.^^ G. Vacca 
{ihid., p. 384) noted the absence of contradiction as (i) leads to equation (1) 
only if a: be a prime. 
C. Moreau®^ treated the equation, of type (1), 
While he used the language of extracting the square root of X = x*+ . . . 
written to the base x, he in effect put X={x^-\-a)^, 0<a<x. Then a^ = 
x+1, 2ax^ = x^-\-x^, whence 2a = a^, a = 2, x = 3, y = ll. 
E. Lucas®^ stated that {x^-\-y^)/{x+y) =^ has the solutions 
(3,-1, 11), (8, 11, 101), (123, 35, 13361),. . . 
Moret-Blanc^^ gave also the solutions (0, 1, 1), (1, 1, 1). 
E. Landau^^ proved that the equation 
— T=y^ 
x — 1 
is impossible in integers (aside from x = 0, ?/ = =fc 1) for an infinitude of values 
of n, viz., for all n's divisible by 3 such that the odd prime factors of n/3, if 
any, are all of the form 6y— 1 (the least such n being 6). For, setting 
n = 3m, we see that y^ is the product of x^+x+1 and F = x^"'~^-{- . . . +a:^+l. 
These two factors are relatively prime since x^=l gives F=m (mod x^+ 
a: + 1 ) . Hence x^+x+lis Si square, which is impossible for a; f^ since it lies 
between x^ and (a:+l)^. 
Brocard^^ had noted the solution a: = 1, y=m, if n = mP. 
A. Gerardin^^ obtained six new solutions of problem (i) : 
a: = 2.47.193.239.701, 2/ = 2^3l5M3M7.97.149. 
x = 2.5.23.41.83.239, y = 2\S\5\7.1S\29.53. 
x = 3.13.23.47.83.239, y = 2^^3^517.13117.53. 
X = 2.3.13.23.83.193.701, y = 2^3^5^7.13.17.53.97.149. 
a; = 3.5.13.41.193.239.701, 2/ = 2^3l5l7.13M7.29.97.149. 
a; = 2.5.13.43.191.239.307, ?/ = 2i^32.5MlM7.29.37.53.113.197.241.257. 
Also <t{N^)=S^ for Ar = 3-7-ll-29-37, ^ = 3-7-13-19-67. 
"Nouv. Ann. Math., (3), 3, 1884, 538-9. 
'"Oeuvres, 2, 434, letter to Carcavi, Aug., 1659. 
"Nouv. Ann. Math., (3), 2, 1883, 30&-10. Cf. Chapter on Diophantine Equations of order 2. 
"L'intermMiaire des math., 7, 1900, 31, 84. 
"Nouv. Ann. Math., (2), 14, 1875, 335. 
»Ibid., 509. 
«76id., (2), 20, 1881, 150. 
"L'interm^diaire des math., 8, 1901, 149-150. 
"Ibid., 22, 1915, 111-4, 127. 
