416 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



Since the values given by n = and n = 1 are rational, those given by any n 

 are rational. Euler stated that if we employ only even values of n, we 

 obtain integral values for x, y. Cayley 152 gave a generalization to several 

 variables. 



A. M. Legendre 83 reduced ay z +byz+cz~-{-dy+fz+g = Q to 



(9) ayt+by&i+Gi^&D, D = fc 2 -4ac>0, -A = a/ 2 -6d/+cd 2 +0D, 



by setting y=(yi+a)/D, z = (z 1 +f3)/D, a = 2cd-fb, P = 2af-bd. If (9) 

 has a solution, it has an infinitude of solutions given by 



(10) yi = yF+8G, z^eF+tG, F+G VD = (<+ ^VD), 



where <f>, \p give the least solution of < 2 Dip = l. It is a question of 

 the values of n for which y and z are integers. Since 



F=(j> n , G=n(j> n - l t, 2 =1 (modD), 

 we see that the expressions for y, z are integers if and only if 



==0 (modD), 

 (modD). 



In either case the resulting values of n are said to be of the form V-\-Dk 

 [denied by Dujardin 84 ], where k is arbitrary, so that there is an infinitude 

 of values n. It remains to solve the problem: if F and G are given by 

 (10s) and if < 2 D\f/ 2 = 1, find all values of n such that XF+juG+J' is divisible 

 by a prime not dividing D\j/. For this, the method of Lagrange 79 is given. 



Dujardin 84 agreed with the statements in the preceding paper down to 

 the erroneous one that the values of n are of the form V+Dk. But the 

 quantities 5, f are divisible by D and the conditions marked (n = 2m) and 

 (n= 2m-\- 1) are satisfied only if the coefficients of the unknowns are relatively 

 prime. Hence a+T, /3+e must be divisible by D if n is even, and 7<+a, 

 0+j8 if n is odd; then the conditions cited are satisfied for all values of m. 

 The correct conclusion is therefore that n varies according to an arithmetical 

 progression of difference 2 (not D). The latter result is said to follow also 

 from the law of recurrence between three consecutive solutions of (9), 

 which leads also to the following rule. Given two consecutive solutions 

 y't, z'i (i = l, 2) of (9) ; then if no one of the systems y\+a, zl+p (i=l, 2) is 

 divisible by D, there is no solution in integers; but if one of the latter 

 systems is divisible by D, then to every system of the same parity as it 

 there corresponds a solution of the proposed equation. 



C. F. Gauss 85 treated the integral solutions of 



(11) ax*+2bxy+cy 2 +2dx+2ey+f=0. 

 Set 



83 Th6orie des nombres, 1798, 451-7; ed. 3, 1830, II, 105-112, No. 439. 

 84 Comptes Rendus Paris, 119, 1894, 843, 934. Reprinted, Sphinx-Oedipe, 4, 1909, 45-7. 

 86 Disquisitiones Arithmeticae, 1801, arts. 216-221; Werke, I, 1863, 215. German transl. by 

 H. Maser, 1889, pp. 205-211. 



