634 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XXII 



quadratic equation in u is satisfied. Equating to zero the coefficient of u z 

 by choice of S, we get u rationally. He deduced Realis' 85 result. 



A. Cunningham" listed all a 4 +6 4 = mc 2 <10 7 , l+?/ 4 = wc 2 , 7/<1000. 



E. Fauquembergue 100 proved Lucas' 73 result that 3, 1, 2 is the only set 

 of solutions of x* ?/ 4 = 52 4 . 



W. Mantel 101 proved by descent that z 4 +2 n 2/ 4= M 2 unless n=3 (mod 4). 



H. C. Pocklington 102 proved by descent the impossibility of 



where p is a prime 8m+3, and indicated (p. 119) the solution of 



R. D. Carmichael 103 treated x*-\-my*=nz 2 . If there is a solution, there is 

 an integer p such that wp 2 = s 4 +m 4 . Hence we are led to the equation 



(7) x*my*=(s 4 +m^z\ 



A special solution, other than the evident one x = s, y = t, = 1, is obtained 

 by setting z = p 2 +mq 2 . Then (7) is satisfied if 



x z = S 2(p 2 _ mq ^ _j_ 2mt 2 pq, y 2 = t 2 (p 2 - mq 2 } - 2s 2 pq. 

 A solution of this double equation is found by the method of Fermat : 

 x = sp-2s(s 8 -m 2 t 8 '), y 



By the method of infinite descent, he proved (pp. 19-21) that there is 

 no set of integers, all different from zero, satisfying either of the equations 

 x 4 4?/ 4 = 2 2 . Hence the area of no rational right triangle is the double 

 of a square ; this implies that x*+y 4 = z 2 has no integral solutions all different 

 from zero. 



A. Gerardin 104 explained three methods to obtain the complete solution 

 of a 4 +fo/ 4 = cz 2 , given one solution. 



A. Auric 105 solved ax 4 -\-by* = cd-z 2 by eliminating z between it and the 

 auxiliary equation mx-+ny 2 = cdz and making the discriminant of the 

 eliminant a square. 



M. Rignaux 106 obtained an infinitude of solutions of z 4 y* = az~, given 

 one solution. *E. Haentzschel 106a discussed (1). 



ax* +by* -\-dx~y 2 MADE A SQUARE. 



L. Euler 107 noted that in making F^x*+kx 2 y 2 +y 4 a square there is a 

 lack of generality in assuming that F is the square of x 2 -\-y^pfq or 



99 L'interm6diaire des math., 18, 1911, 45-6. 



100 L'interm6diaire des math., 19, 1912, 281-3. 



101 Wiskundige Opgaven, 11, 1912-4, 491-5. 

 102 Proc. Cambridge Phil. Soc., 17, 1914, 110. 



103 Diophantine Analysis, 1915, 77-79. 



104 L'interme'diaire des math., 22, 1915, 149-161. 



105 Ibid., 23, 1916, 7-8. 



106 Ibid., 24, 1917, 14. 



1060 Sitzungsber. Berlin Math. Gesell., 16, 1917, 9-16. 



107 Nova Acta Acad. Petrop., 10, ad annum 1792, 1797 (1777), 27; Comm. Arith., II, 183. 



