CHAP. XXII] ax 4 +by*+dx 2 y z MADE A SQUARE. 635 



x*+xyp/qy 2 . By a certain device he was led to the case k =fx 2 +2 VI +fy 2 

 in which F is the square of y 2 +x 2 Vl+/?/ 2 . For 1 </< 100, he gave the least 

 integer y for which the radical is rational. For half of the positive values 

 of k < 100 and for 30 negative values numerically < 100, tables show values 

 of x : y for which F is a square. 



Euler 108 resumed the solution of x 4 +mx 2 y 2 -{-y' l = z 2 . The resulting frac- 

 tion for m can be given an integral form by use of a rational number a for 

 which z = ax 2 y 2 -(x 2 y 2 ). Then -b2 = (az 2 T2)(m/ 2 -2). We may set 

 x = pq, y = rs,a = bl (p V 2 ) , where p, q are relatively prime, likewise r, s. Then 



m2 = (6g 2z F2r 2 )(&s 2 -2p 2 )/(p 2 r 2 ). 



Set 6s 2 -2p 2 = cr 2 , bs 2 +cr 2 = 2n, 6c = X. Then n 2 -p 4 = \y 2 , where y 2 is the 

 largest square dividing n 2 p 4 . Thus m=(\q 2 ^2n)/p 2 . Conversely, for 

 assigned values of p, n, q, the integer x = pq and the largest square y 2 dividing 

 n 2 p 4 are solutions of the proposed equation with the preceding value of m. 

 In fact, 



Euler gave tables of solutions with a slightly changed notation. In con- 

 clusion (p. 498), he gave a more elegant method for the case ra = Xf 2 a, 

 where a 2 - 4 = X/3 2 . Then x = j3, y = 2f , z = /3 2 2af 2 are solutions. Starting 

 with two sets of solutions a, /3 and 2, of the Pell equation, he derived the 

 solution 



Since 0fc = l, A 2 -XJS 2 = 4. Thus for m = X/ 2 A (/ arbitrary), we get the 

 solutions x=B, y = 2f of the quartic equation. 



Euler 109 proved that ra 4 +14ra 2 ?i 2 +7i 4 is not a square if m and n are 

 relatively prime and m is even and n odd (excluding m = 0, n = l), or if m 

 and n are both odd (excluding m = n = l). The question was reduced to 

 one on a 2 +3/3 2 = D. By setting x = m 2 n 2 , y = 2mn, we see that x 2 -\-y 2 

 and 2 +4?/ 2 are not both squares for x odd, y even 4=0. Another corollary 

 is that p(p+<?)(p+2g)(p+3g) =(= D, so that four squares cannot be in arith- 

 metical progression. Another corollary is p 4 p 2 q 2 +q*3= D if p 2 4=<? 2 =l=0, 

 and is derived by setting p = ra+n, q = m n f or p and g odd, and p+<? = 7w, 

 p q = n when one of p, q is even and the other odd. 



Euler 110 elsewhere stated that x 4 z 2 +l=f=n if a; 2 4=1 -or 0. This was 

 proved by the editor of the 1810 English edition, p. 112, by showing in 

 the Appendix that p 2 q 2 and p 2 +3<? 2 are not both squares. 



C. F. Kausler 111 wrote z = x{y in Euler's 107 quartic F. The problem is 

 now to make z*+kz z +l D, or as a generalization f 2 +bZ+eZ 2 = P 2 , Z = z z . 



108 Mem. Acad. Sc. St. Petersb., 7, armies 1815-6, 1820 (1782), p. 10; Comm. Arith., 



II, 492. For misprints and errata see Cunningham. 136 



109 Me"m. Acad. Sc. St. Petersbourg, 8, annees 1817-18 (1780), 3; Comm. Arith., II, 411-13. 



Same results by V. A. Lebesgue, Nouv. Ann. Math., (2), 2, 1863, 68-77. 



110 Algebra, 2, 1770, art. 142; 2, 1774, p. 169; Opera Omnia, (1), I, 403. 

 m Nova Acta Acad. Petrop., 13, ad annos 1795-6, Mem., pp. 205-36. 



