History Of The Theory Of Numbers - I

CHAP, ii] PKOBLEMS ON SUMS OF DIVISOHS. 57
E. Fauquembergue,59 after remarking that (1) is equivalent to the system (2), cited Fermat's60 assertion that the first two equations (2) hold only for x=7 [aside from the evident solutions x = =»=l, 0], which has been proved by Genocchi.61
H. Brocard62 thought that Fermat's assertion that 73 is not the only solution of problem (i) implied a contradiction with Genocchi.61 G. Vacca (ibid., p. 384) noted the absence of contradiction as (i) leads to equation (1) only if 2 be a prime.
C. Moreau63 treated the equation, of type (1),
While he used the language of extracting the square root of .XT=x4-4-written to the base x, he in effect put X = (x2+a)2, 0<a<x. Then a2 = zH-1, 2az2 = x3-fs2, whence 2a = a2, a=2, x = 3, y==lL
E. Lucas64 stated that (x5+y5)/(x+y) = z2 has the solutions
(3,-l, 11), (8, 11, 101), (123, 35, 13361),. . .
Moret-Blanc65 gave also the solutions (0, 1, 1), (1, 1, 1). E. Landau66 proved that the equation
— =</2 x — 1
is impossible in integers (aside from x = 0, y = =*= 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 6v— 1 (the least such n being 6). For, setting n = 3m, we see that y2 is the product of x2-hx+l and jF = x3m~3+ . . . -fa;3-f 1. These two factors are relatively prime since x3==l gives F=m (mod x2+ x+ 1) . Hence x2+£-h 1 is a square, which is impossible for x ^ 0 since it lies between x2 and (x+1)2.
Brocard62 had noted the solution x = l, y=m. if n==m2. A. G6rardin67 obtained six new solutions of problem (i) :
x = 2.47.193.239.701, 7/ = 27.33.53.133.17.97.149.
x = 2.5.23.41 .83.239, y = 28.33.52.7.133.29.53.
x = 3.13.23.47.83.239, 2/ = 2n.32.53.7.133.17.53.
x = 2.3. 13.23.83.193.701, 2/ = 28.33.54.7. 13.17.53.97.149.
x = 3.5. 13.41. 193.239.701, 2/ = 29.33.53.7.133.17.29.97.149.
x = 2.5.13.43.191.239.307, y = 213.32.58.H2.17.29.37.53.113.197.241.257.
Also er(N2) = S2 for Ar = 3-7-ll-29-37, 5 = 3-7-13-19-67.
B9Nouv. Ann. Math., (3), 3, 1884, 538-9.
800euvrcs, 2, 434, letter to Carcavi, Aug., 1659.
fllNouv. Ann. Math., (3), 2, 1883, 306-10. Cf. Chapter on Diophantine Equations of order 2.
«L'interm6diaire dcs math., 7, 1900, 31, 84.
«Nouv. Ann. Math., (2), 14, 1875, 335.
«/6id., 509.
"Ibid., (2), 20, 1881, 150.
ML'interm6diaire des math., 8, 1901, 149-150.
nrhi/i 99 IQI.K; ni-4. 127.