526 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xix 



s 2 = 4fg. For/=2, g = l, Si=-5 or -3, r = 7, s 2 = 8. We may 

 take as s 3 the second value 3, whence a = 3360, Xi = 74, # 2 = 113, # 3 = 58. 

 A. B. Evans 135 found n numbers a; such that a z i -{-ai + i= D (i = l, , 

 n l), al+ai= D. All but the last condition are satisfied if a r = m 2 +2ma T -i 

 (r = 2, - , ri), whence 



a lt = A+2 n - 1 m n ~ 1 a 1 , A=m?+2m?+2 2 m*-\ ----- \-2 n ~ 2 m n . 

 Then cfi+ai = (2 n ~ l m n - 1 ai+p')- gives oj. D. S. Hart took m = l. 



SQUARE OF EACH OF THREE NUMBERS PLUS PRODUCT OF REMAINING TWO 



A SQUARE. 



L. Euler 136 found solutions of x i -\-yz = p i , y i +xz = q 1 , z-+xy = D. Then 

 p 2 q 2 = (x y)(x-\-y z). Set p q = xy, p-\-q = x-\-yz, whence 



p = x \z. 

 Then x"-\-yz = p 2 gives z = 4(x-\-y). The third condition becomes 



Say i36a (4 x +4y+ s y, Then (x-8s)(y-Ss) =65s 2 . Hence set x - 8s = 5ts/u, 

 y Ss = l3us/t, and to avoid fractions take s = tu. Thus x = 8tu-\-5t 2 , 

 y = 8tu -\-13u 2 . He stated that the same solution is found if we start by 

 taking x=(yz s y )[(2s), the resulting numbers being s(8^+s), t(t 8s), 



To give another method, set x = a?-\-2b } y = W-{-2a, z = ab(ab 4). The 

 first two conditions are satisfied and the third becomes 



which is not discussed. But he noted the solutions x,y,z = 33, 185, 608 and 

 297, 377, 320. Nesselmann, 95a p. 141, treated this quartic with a=l/p. 



J. Lynn 137 took 1, x 1, 4x as the numbers. Then two of the conditions 

 are satisfied and the third is (4o;) 2 +a; 1 = D = (4:rdba) 2 , say, which deter- 

 mines x. 



S. Ward 138 took x = mz, y nz, m-\-n 1/4. Then the first two expressions 

 are squares. The third is a square if l + |n n 2 = D, say (1 en} 2 , which 

 gives n. 



J. H. Drummond 139 took w 2 , mw~, nur as the numbers. Then \-\-mn, 

 wtf+n, ra+n 2 are to be squares. Taking n = \ m, it remains to make 

 \+mn= D, say (1 p?ri)~, which gives m. 



W. Wright 140 made a = x~-\-4:yz, fi = y' 1 -\-4zx, j = z 2 -\-4xy and x-\-y-\-z 

 squares. Take x = y+z. Then j3 and 7 are squares. Take 



136 Math. Quest. Educ. Times, 20, 1874, 86-7. 

 138 Opera postuma, 1, 1862, 258-9 (about 1782). 



13Cu J. Cunliffe, New Series of Math. Repository (ed., T. Leybourn), 2, 1809, I, 172-3, chose 

 it equal to (4ry 4x) 2 to obtain x rationally in terms of y, r. We may give any de- 

 sired value to x + y + z. 



137 C. Mutton's Miscellanea Mathematica, London, 1775, 236-7. 



138 J. R. Young's Algebra, Amer. ed., 1832, 336; 



139 Amer. Math. Monthly, 9, 1902, 232. Misprint of mV for mz 2 . 



140 The Gentleman's Math. Companion, London, 3, No. 15, 1812, 346-7. 



