CHAP. XV] LINEAR FUNCTIONS MADE SQUARES. 455 



S. Ryley 59 found three numbers whose sum, sum of any two, and dif- 

 ference of any two plus unity are squares. Take x-\-y = a 2 , x-\-z = b 2 , 

 y-\-z = l. The remaining conditions reduce to 



2a-+2b 2 +2 = n 2 , a 2 -6 2 +l = r 2 . 



Then 4b- = n-2r-=(n-rm) 2 if n = r(ra 2 +2)/(2m). Take r = 2m. Then 

 4 a 2_ 7l 2_j_2r 2 4= D if m 2 +12= D = (s m) 2 , say. Several used the num- 

 bers 2z 2 +2?/ 2 -i, 2z 2 -2?/ 2 +i 2y 2 -2x 2 +, which satisfy five of the condi- 

 tions. To satisfy 4x 2 4y 2 +l = v 2 , take x+y = v-\-l, 4(x y} = vl. For 

 the resulting x, y, 16(2z 2 +2?/ 2 +^) = 17y 2 +30y+25 = (av-5) 2 , by choice of v. 



Fr . Buchner 60 solved x + 1 = p 2 , x 1 = q 2 by setting p+q = m, p q = 2fm, 

 and sim larly for x+a = p 2 , x b = q 2 . 



T. Baker 61 found four numbers p 2 s, q 2 s, r 2 s, s such that the sum 

 of any two is a square, the difference of any two increased by a square r 2 

 (which is to be found) is a square, and the sum of all four diminished by r 2 

 is a square. Set 2s = r 2 t. We need only make p 2 +t, q 2 +t, r 2 +t, 

 A = p 2 q 2 +r 2 , B = p 2 +q 2 r 2 +t squares. Equate the first three to the 

 squares of p+tfx, q+tfy, r+t/z respectively, thus finding p, q, r. Then 

 A = {p v(q+r] p determines t, and B= D holds if 



_x(y-z) 



x 2 -yz 



\ 



S. Jones 62 found four positive integers x, y, z, y+z x half of whose sum 

 is a square, the sum of any two is a square, the difference of any two in- 

 creased by a given square e 2 is a square, and the sum of the four diminished 

 by e z is a square. Take x-\-y = a 2 , x+z = b 2 , y+z = c 2 , 2y-\-zx = d 2 , 

 y+2zx = e 2 , whence a 2 +e 2 = 6 2 +d 2 = 2c 2 , which are satisfied if 



^ 2 ){c {2p(p 2 -l-)}c 



Then all further conditions are satisfied if 6 2 c 2 +e 2 = D, i. e., 



Now/ is the square of mp 2 2n 2 pvfm -\-rnv 2 if v = 2m?pfn 2 . 



T. Baker 63 found five integers p 2 t, q 2 t, r 2 t, s 2 t, t the sum of any 

 two of which is a square. Set 2t = p-+q 2 m 2 . We need only make . 



r 2 +m 2 q 2 , r 2 -\-m 2 p 2 , s 2 +m 2 q 2 , s 2 -\-m 2 p 2 , A=r 2 +s 2 -{-m 2 p 2 q 2 

 squares. Equate the first four to the squares of 



r-\-x(m g), r-\-z(m p), s-\-y(m q), s-\-w(m p), 



respectively. The resulting relations serve to express r/m, s/m, p/rn, q/m 

 rationally in terms of x, y, z, w. The condition A = D is satisfied by making 

 special assumptions. 



69 Ladies' Diary, 1836, 34-5, Quest. 1586. 



60 Beitrag zur Auflos. Unbest. Aufg. 2 Gr., Progr. Elbing, 1838. 



61 The Gentleman's Diary, or Math. Repository, London, 1838, 88-9, Quest. 1360. 

 2 Ibid., 86-8. 



63 Ibid., 1839, 33-5, Quest. 1385. 



