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



the remainder by 4a+2&. Then an answer is given by 



d, e = (a 2 -n)/d, f=d+e+2a, g = 3e+f+2a. 



Thus for n = 3, take a = 2, 6 = 3. Then d = l!7,e = 7,f= 73/7, g = 253/7. 



L. Euler 106 called the numbers A, B, C, D. Set AB = p 2 -n. Equate 

 the product of AC-\-n and BC+n to (Cx+ri) 2 ; then 



n(A-x? 



~ x 2 -AB ' 



Hence (x 2 AB)fn is to be a square y 2 , whence 



C=(A+B-2x')/y 2 , 

 Similarly, 



D = (A+B-2v)fz 2 , v 2 -nz 2 = p 2 -n. 



In CD+n= D, replace A+B by (A 2 +p 2 -n)/A. Hence 

 A*-2A 3 (x+v)+2A 2 (p 2 -n}+A 2 ny 2 z 2 +4:A 2 xv-2A(p 2 -ri)(x+v) + (p 2 -ri) 2 



is to be a square. It can be made the square of A 2 A(x+v) (p 2 ri) by 

 choice of a rational A. To simplify the formulae, Euler took v= x, 

 z = y. Then the condition becomes 



(A 2 -p 2 +ri) 2 +nA 2 y 2 (y 2 -4:) = D 



and is satisfied if y = 2. It remains only to satisfy p' 2 = x 2 3n. Set 

 p = x-t. Then x = (t 2 +3ri)l(2t},p = (3n - t 2 ) /(2t). To secure homogeneity, 

 set x, p = (3TW 2 * 2 )/(2^) . Then 





2gtu 2ftu Sfgtu 



To find four numbers such that the product of any two increased by the 

 sum of the four is a square, we have only to take mA, , mD, where 

 m=(A-\-B+C-\-D)[n, while A, , D, n are the numbers given by the 

 preceding solution. Euler gave two solutions in integers: 15, 175, 310, 

 475 and 36, 96, 264, 504. Since n may be negative, we obtain four numbers 

 the product of any two of which decreased by the sum of the four is a 

 square. A solution in integers is 8, 24, 44, 80. 



E. Bahier, 85 pp. 199-208, employed the numbers of Saunderson, 78 

 taking his two values of z as two of the four numbers. There remains 

 only the condition (r+s) 2 3a= D, which is satisfied by expressing 3a as a 

 difference of two squares. 



OTHER PRODUCTS OF NUMBERS IN PAIRS INCREASED BY LINEAR FUNCTIONS 



MADE SQUARES. 



J. Collins 107 made the six functions xyv, xzv, yzv squares, where 

 v = x j ry+z. Take xyv=(ts} 2 , xzv=(rq) 2 , yzv=(pri) 2 , and (1) 

 i v = ts = rq = pn. Th<mxy = t 2 +s z ,xz = r*+q' i ,yz = p 2 +n 2 . T&ket=(a?-b 2 )g, 



106 Comm. Arith., II, 582-5 (posth. paper); Opera postuma, 1, 1862, 134-7; Algebra, 2, 



1770, arts. 233-4; 2, 1774, pp. 306-14; Opera Omnia, (1), I, 465-9. 

 107 The Gentleman's Math. Companion, London, 2, No. 10, 1807, 66-7. 



