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

If X be chosen so that 0(X), {a02(X)+6)1/2 and {c<£2(X)+d}1/2 take rational values, rational solutions of the pair of equations ax2+b= D, cx2+d= D, are Zi = <£(X), o:2 = 0(2X), Z3 = <£(X+2X), • • •• In order that there be an infinity of solutions, it is necessary that the integral (2) have an irrational ratio to the same integral extended from x to oo .

M. Rignaux131 proved that 2y2+l and 3i/2+l are both squares only when

MISCELLANEOUS PAIRS OF QUADRATIC FUNCTIONS MADE SQUARES.

Diophantus, II, 31, made xy±(x+y} squares. Since 22-j-32zb2-2-3 is a square, take zy=(22-f 32)x2, x+y^2>2-3x2, whence y = 13x, 14o; = 12z2.

Paul Halcke132 gave three ways of solving the problem.

L. Aubry, Welsch and E. Fauquembergue133 proved that the problem is impossible in integers.

Diophantus, II, 26, found two numbers (I2xz and 7x2) such that the square (16z2) of their sum minus either number gives a square. Hence 19z2 = 4z.

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.

BMscara135 made 7y*+$z2 and 7yz-Sz-+l both squares. Treating the first by the method of the " affected square " (Ch. XII) with Sz2 as the additive quantity and 2z as the least root, we get 7(2js)2-h8;z2=(6z)2. For y=2Zj the second expression becomes 2022-j-l and is a square for 2 = 2 or 36.

W. Emerson136 made xy+x and xy-\-y squares.

Fr. Buchner137 made xy—x and xy~ y squares by taking y — p-x+\.

'

S. Tebay138 made x2+cx7/-f7/2±a squares. Let x2+cxy+y-+a = determine y. Then x*+cxy+y*— a= D if x4+ • • - — (x2— cpx+q)2, which gives x.

Several139 proved that P+Q = E2, P2+Q2=S2 imply that P3+Q3 is a sum of two squares :

Also PQ is divisible by 12. To find140 all integral solutions P, Q, set Q = Pq[p. Then P+Q=.R* gives P, while P2+Q2 = s2P2 if p2+ff2 = p2s2 and hence if p = m2 — n2, q = 2mn.

131 L'interm&liaire des math., 25, 1918, 94-5.

m Deliciae Mathematicae, oder Math. Sinnen-Confect, Hamburg, 1719, 245-6.

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

134Rahn'8 Algebra, Zurich, 1659, 110. An Introduction to Algebra, transl. by T. Brancker

. . . augmented by D. P[ell], London, 1668, 100. m Vija-ganita, § 187; Colebrooke,107 p. 252. 138 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 7toU, 60, 1894, 128. Cf , Teilhet389 of Ch. XXI.