452 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xv 



Take c = 2 mn (r 2 + 2s 2 ) - (m 2 - n 2 ) (r 2 - 2s 2 ) . Then c- = b 2 - 2e 2 gives 



n : m = 12r 2 s 2 -r 4 -4s 4 : 8s 4 -2r 4 . 



Cunliffe 42 treated the last problem and Prob. 8 : Divide n into four parts 

 the difference of any two parts being a square. Also Prob. 9: Find four 

 numbers whose sum and sums by twos are squares. 



R. Adrain 43 made two or three linear functions rational squares as had 

 Lagrange. 29 



Several 44 found two numbers such that if unity be added to each and to 

 their sum and difference, the sums are squares. The numbers x 2 2x 

 answer the first two conditions. Then 4x+l = D =p 2 , 2z 2 +l = D. Take 

 p = r+l. Then 16 (2z 2 +l) = (r 2 +4) 2 if r = -8, whence the numbers are 120, 

 168. 



S. Johnson 45 found integers x, y, z, v such that their sum and the sum of 

 any two are squares and 2(v+x-\-y) = D. Set x+y+z+v = a 2 , x+z = b 2 , 

 = c 2 , x-\-y = d 2 . Thus 2z = b 2 -\-c 2 d 2 . Then v+x = a 2 y z = a 2 c 2 , 

 = a 2 b 2 , v-\-z = a 2 d 2 must be squares. Set a 2 c 2 = e 2 , a 2 d 2 =f 2 , 

 c=rp-f, e = sp+d. Then c 2 +e 2 = cZ 2 +/ 2 gives p = (2r/-2sd)/(r 2 +s 2 ). To 

 obtain integers , take / = (n 2 m 2 ) (r 2 + s 2 ) , d = 2mn (r 2 + s 2 ) . Then 



e = (r 2 - s 2 ) 2mn + (n 2 - m 2 ) 2rs, c = (r 2 - s 2 ) (n 2 - m 2 ) - 2nm 2rs. 



By a 2 = d 2 +/ 2 , a=(n 2 +m 2 )(r 2 +s 2 ). Thus a 2 -6 2 =D if fe = (n 2 +m 2 ) -2rs. 

 Finally, 



if n : m = 2rs(r 2 +s 2 ) 2 : s 6 r 2 s 4 +s 2 r 4 r 6 . 



Johnson 46 used the same methods to find x, y, z, v whose sum is a 

 square and difference of any two is a square. J. Cunliffe took v x = c 2 , 

 vy = b z , vz = a?, v-{-x-\-y-}-z = n; it remains to make xy = b 2 c z , 

 xz = a?c 2 ,yz = a 2 b 2 squares. Hence we desire three squares a 2 , 6 2 , c 2 

 the difference of any two of which is a square. This is stated to be true if 

 a 2 = 485809, 6 2 = 451584, c 2 = 462400. 



The problem 46 " to find three numbers in A. P., the sum of any two of 

 which exceeds the remaining one by a square, reduces to x 2 -\-z 2 = 2y 2 (Ch. 

 XIV). 



J. Cunliffe 466 found two rational numbers (x 2 -{- n and y 2 +ri) such that 

 each and their sum and their difference exceed a given number n by squares. 

 The condition x 2 -\-y 2J t~n= D = (x+f) 2 gives x in terms of y, v, n. Then 

 x 2 y 2 n=n = (n v 2 y 2 ) 2 /(4v 2 ) if n 2 2nv 2 = \3 = (rv n) 2 , which deter- 

 mines v. 



42 The Math. Repository (ed., Leybourn), London, 3, 1804, 97-106. 



The Math. Correspondent, New York, 1, 1804, 237-241; 2, 1807, 7-11. 



" Ladies' Diary, 1804, pp. 38-9, Quest. 1111; Leybourn's Math. Quest. L. D., 4, 1817, 23. 



46 The Gentleman's Math. Companion, London, 2, No. 8, 1805, 46-8. 



IUd. t 2, No. 9, 1806, 35-6. 



460 New Series of Math. Repository (ed., Leybourn), 1, 1806, I, 7-10. 



466 Ibid., 2, 1809, I, 9-11. 



