358 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



and proved that 



O) 2 - z- 1 2 = - a, (v, a} 2 - z(a} 2 = 0, (v, a, 6) 2 - z(a, b} 2 = - 7, 

 (v, a, b, c) 2 z(a, b, c) 2 = 5, (v, a, b, c, d) 2 z(a, b, c, d) 2 = e, 



so that, for example, x 2 zy 2 = y has the solution x = (v, a, 6), 

 y = (a, 6). No one of /3, 7, 5, equals 1 unless the corresponding 

 index is 20. Hence if any period contains the index 2v and if x/y is the 

 convergent defined by this period, we have x 2 zy 2 = 1 or + 1, accord- 

 ing as the number of indices in .the period is odd or even. In the first case, 

 = 2x 2 + 1, 77 = 2xy give a solution of 2 zr? 2 = + 1; or we may take 

 two successive periods and apply the second case. He applied this theory 

 to eight special types of periods, such as v, a, b, b, a, 2v, a, . He recog- 

 nized that we need only use a half period. Thus, for the period just cited, 

 we employ the half period v, a, b of indices and convergents 1/0, v/l, Bjf3, 

 (7/7. Then x 2 zy 2 = 1 for 



x = (v, a, b, b, a) = (a, b)(v, a, b) + (a)(v, a) = yC -f- (3B, 

 y = (a, b, b, a) = (a, b} (a, 6) + (a] (a) = 7 2 + P. 



But if z has the indices v, a, b, c, b, a, 2v, with an even number of terms in 

 the period, we use the half period v, a, b, c and the additional convergent 

 D/8 and find that x- zy 2 = +1 for 



x = (a, b)(v, a, 6, c) + (a)(v, a, b) = yD + |8C, 

 y = (a, 6) (a, b, c) + (a) (a, 6) = yd + Py. 



As equivalent formulas were restated by Tenner, 118 they are often attributed 

 to him rather than to Euler. The formulas are stated in general form by 

 Muir 160a and Konen, 3 pp. 55-6. 



Finally, he tabulated the least solutions of p 2 lq 2 = 1 for each I < 100 

 which is not a square, and for I = 103, 109, 113, 157, 367 [errata for I = 33, 

 83, 85, Cunningham 309 ]. 



J. L. Lagrange 74 gave the first proof that x 2 ay 2 = 1 has integral 

 solutions with y =1= 0, if a is any integer not a square. He noted that 

 Wallis 61 committed a petitio principii in attempting a proof, while the 

 method of solution explained by Wallis 49 is tentative and not shown to 

 succeed. Lagrange started with the continued fraction 



and its successive convergents m/n, M/N, m'fn', M'/N', . Taking 

 fo y} = (M, N), (M', N'), - -, we always obtain positive values < 2M/N 

 for x 2 ay 2 . Hence an infinitude of these values are identical. Let 

 (x, y), (x', y'), (x", y"), be an infinitude of pairs of integers for which 

 x 2 ay 2 has the same value R. First, let R, a be relatively prime. By 

 multiplication and by elimination of a, 



(A) R 2 = (xx r ayy'Y ~ - a(xy' yx') 2 , 



(B) R(y' 2 - y 2 ) = x 2 y' 2 - y 2 x' 2 . 



