CHAP. XXII] QUARTIC FUNCTION MADE A SQUARE. 639 



Cases in which ax*+dx*y 2 +by*= D is impossible were noted by 

 Lebesgue, 30 - 37 Genocchi 85 - 93 and Pepin 98 of Ch. XXVI; by Desboves 188 of 

 this Chapter. Solvable cases by Pepin 132 of Ch. I, Haentzschel 143 of Ch. V. 



QUARTIC FUNCTION MADE A SQUARE. 

 Fermat 139 sought rational values of x for which 

 (1) f(x) = a+bx+cx 2 +dx 3 +ex* 



shall equal the square of a rational number, where a, , e are integers. 

 The case in which a or e is the square of an integer is quite simple. For 

 a = a 2 , the first three terms of f(x] are identical with those of the square of 



Comparing the terms with the factor x 3 , we obtain 



8o: 2 [6(4a 2 c-6 2 )-8a 4 (r| 

 64 6 e-(4a 2 c-6 2 ) 2 



Hence from a particular solution /() = a 2 , we may obtain new solutions 

 since /(+) =cr+bx-\- +&r 4 falls under the last case. 



The same special cases were treated similarly by L. Euler 140 and 

 A. M. Legendre. 141 



T. F. de Lagny 142 made o; 4 -10x 3 +26x 2 -7o:+9 the square of x*- 

 for x = 23/5. 



L. Euler 143 treated in a posthumous paper the equation 



a 2 z 4 + 2abx*y + cx~y 2 + 2bdxy* + dY = D 

 Set c-b*-2ad = mn. Then 



(2) (ax*+bxy+dy~}~-{-mnx' 2 y' 1=z2 - 



This is satisfied if 



ax 2 -\-bxy-{-dy 2 = \(mp 2 ng 2 ), xy = 2\pq, z 

 Admitting fractional solutions, we may set y = 1. Then 



For a fixed X, let p and q be given solutions. Let p f be the second root of 

 this quadratic in p, whence 



p' = p 2bq/(4\aq 2 m) . 

 Then p', q' are corresponding values if 



q'= -q-2bp'f(4:\ap' 2 +n'). 



139 Diophanti Alexandria! Arith. Libri Sex . . . Doctrinae Analyticae Inventum Novum; 



Collectum a J. de Billy ex varijs Epistolis quas ad eum . . . misit P. de Fermat, p. 30. 



French transl., Oeuvres de Fermat, 3, 1896, 377-388 (the term x* is omitted on p. 388, 



31). 

 140 Algebra, 2, 1770, Ch. 9, Nos. 128-137; French transl., Lyon, 2, 1774, pp. 153-162. Opera 



Omnia, (1), 1, 1911, 396-400. Sphinx-Oedipe, 1908-9, 67-78. 

 m The"orie des nombres, 1798, 458-9; ed. 3, 2, 1830, 123; Maser, II, 120. 



142 Nouv. Elemens d'Arith. et d'Alg., Paris, 1697, 496. 



143 Me"m. Acad. Sc. St. Petersb., 11, 1830 [1780], 1; Comm. Arith., II, 418. 



