CHAP, xvi] x*+2fxy+hy 2 , x-+2gxy+ky 2 BOTH SQUARES. 481 



Hence the product of r 2 +16rs-s 2 is 252rV; call the factors 14(3p) 2 , 

 2<? 2 and add and subtract. Thus 



For the upper sign, one solution of (1) leads to two new solutions: 



i, S = nt, T-- 



so that the proposed pair has the solutions 



o; = r 2 +s 2 , 4y= r 2 s 2 db, u, v = r 2 s 2 T2rs. 



For the lower sign, the problem is reduced to the earlier case. 

 A. Ge'rardin 120 used the known solution of u 2 +v 2 = 2x 2 : 



u = 2m?-l 2 , v = 2m 2 +l 2 -tti?i, x 

 It remains to make Su 2 7x-= (z+4?/) 2 a square: 



Then a; = 34, y = 15, ^ = 46, v = 14 [Genocchi]. It is stated that we have 

 also y= 32. 



L. Euler 121 solved x 2 +2fxy+hy 2 = P 2 , x~+2gxy+ky- = Q\ Subtract and 

 set P Q=(f-g}y, whence P+Q = 2x+y(h-lc)l(f-g). Squaring and 

 adding, we get 2P 2 +2Q 2 ; equating to the value obtained by adding the 

 proposed equations, we get 



N. Fuss 122 made/i = z 2 +2a:n/+?/ 2 and/ 2 =x 2 +2fon/+?/ 2 both squares, say 

 p 2 and# 2 . Thenp z -q 2 = 2(a-b}xy. Hence x = 4(a+b), y = (a-b)"-4 is a 

 particular solution since 



To find n such that x 2 2nxy+y 2 are squares, say (pq) 2 , we have 



x 2 +y 2 = p z +q z , nxy = pq. 



Set p = axy, q = nfa. Then n 2 = a 2 (x 2 +y 2 ) a 4 x 2 y 2 . 

 A. S. Werebrusow 123 reduced the system 

 ax 2 +2a'xy+a"y 2 = u 2 , (3 



to au*+2buW+cv 4 = z 2 , a = /3 /2 -/5/3 // , 6 = a 



H. B. Mathieu 124 gave the solutions x = 15,y=-8; x = 1768, y = 2415, 

 of z 2 +2/ 2 =D, x 2 +xy+y 2 = D. L. Aubry 125 gave a general discussion. 



Adrain, 113 Genocchi, 119 etc., of Ch. XXII proved that x 2 xy+y 2 are 

 not both squares. 



120 Sphinx-Oedipe, 1906-7, 162; Assoc. frang., 1908, 17. 



121 Opera postuma, I, 1862, 254 (about 1777). Nova Acta Acad. Petrop., 13, 1795-6 (1778), 



45; Comm. Arith., II, 292. 



122 M<m. Acad. Sc. St. Petersbourg, 9, 1824 (1820), 151-160. 



123 Math. Soc. Moscow, 26, 1098, 497-543; Fortschritte, 39, 1908, 259. 

 1M L'intermgdiaire des math., 17, 1910, 219. 



125 Ibid., 283-5. 

 32 



