CHAP, xvi] PAIRS OF QUADRATIC FUNCTIONS MADE SQUARES. 483 



If X be chosen so that <(X), {a^(\}+b} 112 and {c< 2 (X)+d} 1/2 take rational 

 values, rational solutions of the pair of equations ax 2 +b=t3, cx 2 +d=D, 

 are #i = 0(X), z 2 = 0(2X), Xz = <f> (X+2X), . In order that there be an in- 

 finity of solutions, it is necessary that the integral (2) have an irrational 

 ratio to the same integral extended from x to o> . 



M. Rignaux 131 proved that 2y 2 -{-l and 3?/ 2 +l are both squares only when 



MISCELLANEOUS PAIRS OF QUADRATIC FUNCTIONS MADE SQUARES. 



Diophantus, II, 31, made xy(x+y) squares. Since 2 2 +3 2 =fc2-2-3 is 

 a square, take rn/ = (2 2 +3 2 )z 2 , x+y = 2<2-3x 2 , whence y = l3x, I4x = l2x 2 . 



Paul Halcke 132 gave three ways of solving the problem. 



L. Aubry, Welsch and E. Fauquembergue 133 proved that the problem is 

 impossible in integers. 



Diophantus, II, 26, found two numbers (I2x 2 and 7x 2 ) such that the 

 square (16o; 2 ) of their sum minus either number gives a square. Hence 



This problem was treated by J. H. Rahn and J. Pell, 134 and the latter 

 treated (p. 102) the corresponding problem (Diophantus, III, 3) for three 

 numbers. 



Bhdscara 135 made 7y 2 +8z 2 and 7y z Sz 2 -\-l both squares. Treating the 

 first by the method of the " affected square " (Ch. XII) with 8z 2 as the 

 additive quantity and 2z as the least root, we get 7(2z) 2 +Sz 2 =(Qz} 2 . For 

 y = 2z, the second expression becomes 2Qz 2 +1 and is a square for 2 = 2 or 36. 



W. Emerson 136 made xy+x and xy+y squares. 



Fr. Buchner 137 made xyx and xy y squares by taking y = p 2 x-}-l. 

 Then xy y = (px m) z if x = (m-+l)/(2mp p' 2 +l). 



S. Tebay 138 made x*+cxy+y 2 a squares. Let x 2 +cxy+y 2 +a = (y+p} 2 

 determine y. Then x 2 +cxy+y 2 a= D if x*-\ ---- = (x 1 cpx+q) 2 , which 

 gives x. 



Several 139 proved that P+Q = R 2 , P 2 +Q 2 = S 2 imply that P 3 +Q 3 is a 

 sum of two squares : 



Also PQ is divisible by 12. To find 140 all integral solutions P, Q, set 

 Q = Pq/p. Then P+Q = R 2 gives P, while P 2 +Q 2 -s 2 P 2 if p 2 +q 2 = p 2 s 2 and 

 hence if p = m 2 n 2 , q = 2mn. 



131 L'interm^diaire des math., 25, 1918, 94-5. 



132 Deliciae Mathematicae, oder Math. Sinnen-Confect, Hamburg, 1719, 245-G. 



133 L'intermediaire des math., 18, 1911, 71-2, 285-6; 20, 1913, 249. 



134 Rahn's Algebra, Zurich, 1659, 110. An Introduction to Algebra, transl. by T. Brancker 

 . . . augmented by D. P[ell], London, 1668, 100. 



135 Vija-gamta, 187; Colebrooke, 107 p. 252. 



136 A Treatise of Algebra, London, 1764, 1808, p. 379. 



137 Beitrag zur Aufl. unbest. Aufg. 2 Gr., Progr. Elbing, 1838. 



138 Math. Quest. Educ. Times, 44, 1886, 62-3. 



139 Ibid., 54, 1891, 38. 



140 Ibid., 60, 1894, 128. Cf. Teilhet 369 of Ch. XXI. 



