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



Selected values of p give an answer if their sum is 130, as for 2, 5, 26, 32, 

 65, and 2, 13, 26, 40, 49. 



Euler 122 found a, b, c, d so that abcd, ac bd, be ad are squares. Call 

 the first two expressions x 2 , y 2 , and solve for b, c. Take 2x = a-\-d-\-v, 

 2y = a+d v. Then 



(a+dy2(a-d)vl-v 2 [ a~-Qad+d 2 -v"^ 



b, c=- -=r- -, be ad=\ - :- -= - . 



4(a d) 4(ad) 



For v = d = S, a = 24, we get 6 = 21, c=13. 



S. Tebay 123 found four positive integers ai, , 4 such that aic^+ascu, 

 aia 3 +a 2 a 4 , fliCk+c^fls, So f oy are squares. 



A. Gerardin 1230 made xy+zt and xzyt squares by several methods. 



SQUARES INCREASED BY LINEAR FUNCTIONS MADE SQUARES. 



Let <T = Xi+Xz +#3. Diophantus, II, 35, and Bombelli 124 made x]-{-cr 

 a square for i = l, 2, 3. Diophantus, II, 36, made each x] a a square. 

 Diophantus, V, 9, made each x]a a square. Diophantus, III, 1, made 

 each ffxl a square by taking xi=x, x 2 = 2x, o- = 5x 2 , 5 = (2/5) 2 +(ll/5) 2 , 

 x 3 = 2/5, whence x = 17/25. J. Whitley 125 took x\ = x, x 2 = nx, x s = mx, 

 ffxl = a 2 x 2 , which gives x. Then 1+a 2 n 2 and 1+a 2 m 2 are to be 

 squares, which is the case if \rC- = a m. 



Diophantus, IV, 17, made 0:1+^2+0:3, xl+x 2 , o^+^s, ^3+^1 all squares 

 by taking 2 = 4, Xi = x l, 16x 2 +o;3 = (4x+l) 2 , whence Xz = Sx+l. Then 



whence y = 55/52, x = 137/ 2 . 



Fermat 126 suggested that a more elegant solution is obtained by setting 

 x 3 = 4x-\-3, whence 



a "double equation" with squares as constant terms. He stated that a 

 similar device will solve the analogous problem in four or a greater number 

 of unknowns. 



J. Anderson 127 took x\-\-x^- (p Xi} z , xl+Xz=(qx- 2 Y, xl J rXi = (rx^, 

 which give x\, x z , x s . In Szi, equate the coefficient of r 2 to zero, whence 

 5 = 1/4. Other writers gave essentially Diophantus' solution. 



S. Ward 128 took x z = l-2x 1 , (l-2o: 1 ) 2 +X3 = A 2 , l-Xi+x^B 2 . Then 

 A* -B* = 4x*-3xi. Take A+B = 2xi, A-B = 2x l -3/2, whence 5 = 3/4, 

 i = a;3+7/16. Then I6(xl+Xi) = (4x 3 pfq) 2 determines x 3 . 



122 M6m. Acad. Sc. St. Petersb., 5, anno 1812, 1815 (1780), 73 (21); Comm. Arith., II, 



385-91. 



123 Math. Quest. Educ. Times, 52, 1890, 117. 

 1230 L'interme'diaire dcs math., 26, 1919, 17-18. 



124 L'algebra opera di R. Bombelli, Bologna, 1579, 485. 



126 Ladies' Diary, 1807, 37, Q. 1155; Lcybourn's Math. Quest. L. D., 4, 1817, 72-3. 

 Oeuvres, I, 301; French transl., Ill, 249. 



127 The Gentleman's Math. Companion, London, 5, No. 26, 1823, 204-7. 



128 J. R. Young's Algebra, Amer. ed., 1832, 337-8. 



