478 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



AND X + y BOTH SQUARES. 



Diophantus, II, 21, took y = 2x+l. Then x~+y= D. Let 



be the square of 2x 2. Then a: = 3/13. 



Alkarkhi 96 (beginning of eleventh century), after repeating this solution, 

 added the condition x 2 +i/ 2 = D, taking x = 3z, y = 4z. 



Rafael Bombelli 97 set 2/ = 4(z+l). Let x+if=lQx-+33x+lQ be the 

 square of 4z - 6. Then x = 20/81. 



W. Emerson 98 treated the problem. 



L. Euler" set x*+y = (p x) 2 , i/ 2 +z = (g y)~, whence 



_ 

 X= 



Euler had first inserted the value of y from x 2 +y = p z into y*+x, obtaining 

 (p-x^ 2 +x= D, which he stated would be difficult to solve. 



R. Adrain 100 noted that Euler's last condition is satisfied if we take 

 x = 4p' 2 x z . Again, for p+x = v, it becomes 



if 



The equivalent problem x 2 y=E3,y' 2 z= D was solved 101 as by Euler. 99 

 J. W. West 102 noted that Euler's 99 condition is satisfied if p--x- = y, 



x = 2y+l. Solve the quadratic in x obtained by eliminating y. 



C. A. Laisant, 103 after recalling Euler's 99 solution in rational numbers, 



remarked that the system is evidently impossible in positive integers, since 



in 



y = (z-x)x+(z-x)z, x=(t-y)y+(t-y)t, z>x, t>y, 



y>xby the first equation and x>y by the second. Similarly for negative 

 solutions. 



A. Auric 104 noted that Euler's solution is not general, since his problem 

 is equivalent to the solution in integers of the homogeneous system 

 x*+uy = z 2 , ux+y* = P, which can be solved for x, y after giving arbitrary 

 values to z, t, u (by factoring z 2 P). 



L. Aubry 10 '" and G. Quijano 106 proved the impossibility of integral solu- 

 tions. _ 



98 Extrait du Fakhri, French transl. by F. Woepcke, Paris, 1853, 88-9. 

 87 L'algebra opera, Bologna, 1579, 467. 



98 A Treatise of Algebra, London, 1764, 1808, p. 239. 



99 Algebra, 2, 1770, art. 239; French transl., Lyon, 2, 1774, 335-6. Opera Orania, (1), I, 482. 



100 The Math. Correspondent, New York, 2, 1807, 11-13. 



101 The Ladies' and Gentlemen's Diary (ed., M. Nash), N. Y., 2, 1821, 45. 



102 Math. Quest. Educ. Times, 67, 1897, 64. 



103 Nouv. Ann. Math., (4), 15, 1915, 106-8. 



104 Ibid., 280-1. 



105 L'intermeMiaire des math., 22, 1915, 67, simpler on p. 226. 



106 Ibid., 23, 1916, 87-8. 



