630 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP. XXII 



as for the first equation. Next, x 4 2 4M+1 i/ 4 = z 2 implies 



whence x z = a 4 +8j3 4 , xa <2 = 2y 4 , a;To: 2 = 45 4 , 2 = 7 4 -25 4 . For the upper 

 sign we have an equation like the proposed. For the lower sign, there are 

 solutions, as = 7 = 5 = 1. The impossibility of x 4 2 4 " + fy 4 = z 2 ( = or 2) 

 follows from z 2 d=z = 2a 4 , z 2l Fz = 2 3+< J S 4 , z 2 = a 4 +2 2 +'/3 4 . The impossibility of 

 3 4 +2/ 4 = 2 2 * +1 3 2 , rc+2/, follows from (z 4 -?/ 4 ) 2 = 2 4M+2 z 4 -4;ry. 



Lebesgue's 56 results concerning the equations in the last paper have 

 been quoted. Cf. Schopis 19 on x 4 +y 4 ^2z 2 . 



E. Lucas 73 listed and treated the solvable equations 



(1) az 4 +fa/ 4 = cz 2 , 



in which 2 and 3 are the only primes dividing a, b or c, viz., (a, 6, c) = 

 (1, -1, 24), (1, -2, 1), (1, 2, 3), (1, 3, 1), (1, -6, 1), (1, 8, 1), (1, 9, 1), 

 (1, -12, 1), (1, 18, 1), (1, 24, 1), (1, 36, 1), (1, -54, 1), (1, -72, 1), 

 (1, 216, 1), (3, -1, 2), (3, -2, 1), (4, -1, 3), (4, -3, 1), (9, -1, 8), 

 (9, -8, 1), (27, -2, 1). 



T. Pepin 74 stated that there is no rational solution of px 4 36i/ 4 = z 2 if p 

 is a prime of the form a 2 -f 96 2 , and many such theorems with 36 replaced 

 by new numbers, usually by the discriminant of the quadratic form for p. 



Lucas stated and Moret-Blanc 75 proved that x = l, y = Q and # = 3, y = 2 

 are the only integral solutions ^0 of x 4 5y* = l. 



Lucas 750 proved that either of 4^ u 4 = 3w 4 , 9v 4 u 4 = 8w 4 implies 



Pepin 76 noted that necessary conditions for relatively prime integral 

 solutions of Au? = Bx 4 +Cy 4 are that AB, AC and BC be quadratic 

 residues of C, B, A, respectively, and that BC 3 be a biquadratic residue of 

 A. He proved that u? = 3y 4 2x 4 is completely solved by the repeated 

 application of 



x = XV - 3/V, y = X 2 / 2 +2 M V, u = x 2 - 12X M /<7(X 2 / 2 - 2 M V) , 

 where X, /z, /, g are integers relatively prime in pairs such that 



g : = 

 The same analysis gives the complete solution of x 4 6i/ 4 = 



and 



He treated other rare cases in which the complete solution is found: 

 = 8v? and 7x* 2y* = 5u 2 , with the respective auxiliaries x 4 +28y*=z 2 



"Recherches sur Tanalyse ind6termin6e, Moulins, 1873; extract from Bull. Soc. d'Emula- 

 tion du DSpartement de 1' Allier, 12, 1873, 441-532. Bull. Bibl. Storia Sc. Mat. Fis., 

 10, 1877, 239-58. 



74 Comptes Rendus Paris, 78, 1874, 144-8; 88, 1879, 1255; 91, 1880, 100 (reprinted, Sphinx- 

 Oedipe, 5, 1910, 56-7); 94, 1882, 122^. 



76 Nouv. Ann. Math., (2), 14, 1875, 526; 20, 1881, 203-5. 



7s Nouv. Ann. Math. (2), 16, 1877, 415. 



78 Atti Accad. Pont. Nuovi Lincei, 31, 1877-8, 397-427. 



