628 HISTORY OP THE THEORY OF NUMBERS. [CHAP, xxn 



L. Euler 65 proved that 2a 4 26 4 is not a square if a^b by means of the 

 fact that x 4: Fy 4 is not a square. Likewise for 4z 4 2/ 4 , x 4 4?/ 4 , d= (4z 4 2y 4 ) . 

 [Cf. Frenicle, 9 Bendz, 29 Carmichael. 36 ^ He proved that neither wa 4 ra 3 6 4 

 nor its double is a square. Also 66 that a 4 +26 4 =f=n if 64=0. 



Euler 67 treated a+ex 4 =D, supposing known one solution: a+eh 4 = k 2 . 

 $etx = h+y. Then 



a+ex 4 = k 2 +4eh*y+Qeh 2 y 2 +4ehy*+ey 4 

 will be the square of k+2eti*y/k+eh 2 (k 2 +2d)y 2 /k 3 if 



y = 4hk 2 (2a - k 2 ) /(3& 4 - 4a 2 ) . 



By use of the substitution x=h(l+y)l(ly), a+ex 4 becomes a quartic 

 having both the constant term and the coefficient of y 4 squares, and hence 

 is more readily made a square. 



J. L. Lagrange 68 proved that if s 4 +at 4 = u 2 a second set of solutions of 

 x 4 +ay 4 =z 2 is given by 



x = s 4 at 4 , y = 2stu, z = u 4 +4as 4 t 4 . 



To deduce this result, Lagrange made assumptions which he recognized 

 were not necessary ones. Assume that z = m 2 -\-an 2 . Then the given 

 equation is satisfied if y 2 = 2mn, x 2 = m 2 an 2 . The latter holds if m = p 2 + aq 2 , 

 n = 2pq, x = p 2 aq 2 . The resulting expression for y 2 is a square if p = s 2 , 

 q = t 2 , p 2 +aq 2 = u 2 . From the second solution, one deduces similarly a 

 third, etc. But not all sets are necessarily obtained in this way. He 

 remarked that the simplest and most general method for such equations 

 is perhaps that by factors in his Addition IX to Euler's Algebra (Lagrange 163 

 of Ch. XXI). 



A. E. Kramer 69 treated px 4 y 4 = z 2 , where p is an odd prime, and x, y 

 are relatively prime. Let p = 7i 2 +w 2 . Then 



(y 2 +mx 2 ') (y 2 mx 2 ) = (nx 2 -\-z) (nx 2 z) . 



He took m = r 2 . First, let one of y, r be odd and the other even, so that x is 

 even. Set y 2 +r 2 x 2 = ab, nx 2 +z = ac, where b, c are relatively prime. Then 

 the long equation gives y 2 r 2 x 2 = dc, nx-z = db. Then a, b, c, d are odd 

 and a, d are relatively prime. Since a = na r 2 d, (3=r 2 a-\-nd have no 

 common factor except possibly p, while ba = c(3, we have a = sc, (3 = sb, 

 where s = 1 or p. Let e be the g. c. d. of d and y-\-rx; h that of afs and 

 yrx. Since y 2 r 2 x 2 -dafs, we get y-\-rx = ef, yrx = gh, d = eg, a{s=fh, 

 where /, g are relatively prime, as also e, h. Substituting the values of y, rx, 



65 Comm. Acad. Petrop., 10, 1747 (1738), 125; Comm. Arith., I, 28; Opera Omnia, (1), II, 47. 



Algebra, St. Petersburg, 2, 1770, arts. 209-10; French transl., Lyon, 2, 1774, 254-263. 

 Opera Omnia, (1), 1, 442-3. 



66 The proof in his Algebra is the shorter. The latter was reproduced by A. M. Legendre, 



Theorie des nombres, 1798, p. 405; Maser, II, 7; E. Waring, Medit. Algebr., ed. 3, 1782, 



373-4. 

 87 Algebra, St. Petersburg, 2, 1770, Arts. 138-9; French transl., 2, 1774, pp. 162-7; Opera 



Omnia, (1), I, 400-2. 

 "Nouv. Mem. Acad. Sc. Berlin, annde 1777, 1779, 151; Oeuvres, IV, 395. Reproduced 



by E. Waring, Meditationes Algebraicae, ed. 3, 1782, 371. 

 89 De quibusdam aequationibus indeter. quarti gradus, Disa., Berlin, 1839. 



