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



T. Pepin 111 found that all rational solutions are given by 

 dx = m 2 p-\-n 2 q, dy = kmnst, dz = 2mn(s 2 -}-2t 2 '), du = 2mn(s' 2 2 2 ), 



where d = m 2 p n 2 q, while m, n are relatively prime, also s, t. Further, 

 s*+4:t 4 = pq. To obtain integral solutions, take 

 p = l } d=l, 



Various writers 112 gave solutions. 



J. H. Drummond 113 took 

 m = 2n 2 , whence x = 2n 2 , y = 2n. 



E. B. Escott 114 asked if x 2 -{-y 2 1, x 1 y~-\-l are squares when 

 x^y, for integral values other than z = 13, y = ll', in other words, if 

 4wn(m 2 n 2 ) + l = D (mn^Q, m^ri) has a solution other than m = 3, n = 2. 

 Several replies 115 show there is an infinitude of solutions. 



R. P. Paranjpye 116 gave Bhascara's 107 third solution. Suppose in 

 2y- = z 2 t 2 that the common factor of y, 2, is a square. Since the difference 

 of two squares is divisible by 8, we may set z+t=4 2 , z t = 2ri 2 , y = 2^rj. 

 Then X 2 = l-y 2 + } z 2 = 4:+r) 4 +l. Assume that ?? 4 = 4 2 , whence = 



x 2 +2fxy+hy 2 , x-+2gxy+ky- BOTH SQUARES. 



Beha-Eddin 32 listed as the last of seven problems remaining unsolved 

 from former times that to make x 2 (x-\-2') both squares. His translator, 

 Nesselmann (pp. 72-3), discussed the problem. 



A. Marre 117 found only the solution x= 17/16 and concluded that 

 the problem is impossible in positive integers. 



A. Genocchi 118 called the squares (p-\-q)~ and (p q} 2 , whence X 2 = p 2 +q 2 , 

 x+2 = 2pq. By eliminating x, (4p 2 l)q 2 8pq(p 2 4:)=Q. By taking 

 the first or third coefficient zero, we get x= 2, 17/16, 34/15. 



E. Lucas 119 solved completely the corresponding homogeneous equations 



where x, y, u, v may be assumed relatively prime. Adding, we see that the 

 sum of the squares of (uv)/2 is x 2 , whence 



=r 2 s 2 , %(u v) 

 Substitute the resulting values of u, v, x into the equation obtained by 

 subtracting the proposed equations, we get 2y~-\-xy = ^rs(r 2 s 2 ), whence 

 y=l(xi), where 

 (1) (r 2 +s 2 ) 2 +32rs(r 2 -s 2 ) =t 2 . 



111 Nouv. Ann. Math., (2), 14, 1875, 63. 



112 Math. Visitor, 2, 1887, 6G-70. 



113 Amer. Math. Monthly, 9, 1902, 232. 



114 L'interm<diaire cles math., 12, 1905, 76. 



116 Ibid., 207-211; 13, 1906, 25. Cf. Zerr 50 of Ch. XIX. 



116 Jour. Indian Math. Club, Madras, 1, 1909, 188-9. 



117 Nouv. Ann. Math., 5, 1846, 323. 



118 Annali di Sc. Mat. e Fis., 6, 1855, 132, 303-4. 



119 Nouv. Ann. Math., (2), 15, 1876, 359-365. Same in Bull. Bibl. Storia Sc. Mat., 10, 1877, 



186-191. 



