CHAP, xxii] A 4 +JS 4 = C 4 + 4 . 645 



Euler 167 treated a 4 6 4 = c 4 d 4 by setting 



( a 2 +fe2 ) p== ( c *-d 2 )q, (a 2 -b 2 )q = (c 2 +d z )p. 



Multiply the first by p, the second by q, add and subtract. Let q 2 p 2 = s z . 



Then 



(2) 6 2 s 2 = a 2 (p 2 +g 2 ) -2c 2 pq, 2d 2 pg = aV-6 2 (p 2 +g 2 ). 



In (2i) take bs = a(q p)+2p(a c)x, whence 



a : c = 2px 2 -\-q : 2px 2 +2(qp')x q. 



Taking a and c equal to these expressions, and multiplying (2 2 ) by s 2 /(2pq), 

 we find that 



d 2 s z = q 2 (q-p) 2 -4q(q-p)(q 2 +p 2 ')x+2(q 2 -p 2 ) 2 x 2 



which is readily made a square since the first and last coefficients are squares. 

 For p = 3, q = 5, we have s = 4 and 



(3) d 2 = - 



If we seek to make three terms of d 2 identical with those of the square of 

 5/2 - I7x+ax 2 or of a+ 17x+3x z , we find that c 4 = a 4 . But 



-7= D =f 2 , 



for 2 = a+|Sz, z = (5f)/e; also for 2 = z(ez+5), z = a/(|8d=f). For the 

 special form (3) we therefore get z = 15, 11/3, 1/18 or 5/22, each leading 

 to a permutation of the same values 



a = 542, 6 = 359, c = 514, d = 103. 

 Euler 168 treated the following generalization of (1) : 



pq (rap 2 + nq 2 ) = rs (mr 2 + ns 2 ) . 

 Set q = ra, s = pb. Then p 2 : r 2 = na?mb : ntfma. Set 



a = 6(1+2), a = nb 2 l(nb 2 -7n), /3 = a-l. 

 Then 



We may make (7(1 &z) D by the usual methods for quartics. But we 

 obtain much simpler solutions by making C{(lpz) = (l+dz) 2 , viz., 



Taking 2d=3a+/3, we get 2= -3/(4a+4/3d 2 ), p/r=l+dz. 



For m = n = l, 6 = 3, we get a = 9/8, /3 = l/8, c^ = 7/4, 2= -96/193, 

 p/r = 25/193, and obtain the solution p = 25, r = 193, g = 291, s = 75 of (1), 



167 M6m. Acad. Sc. St. Petersb., 11, 1830 (1780), 49; Comm. Arith., II, 450. Euler wrote 



c 2 +d 2 in his second equation and c 2 d 2 in his third. His further formulas require that 

 d 2 be replaced by d 2 , which would invalidate the conclusions. In the present report, 

 d 2 has been replaced by d 2 at the outset, so that the remaining developments become 

 correct as they stood. 



168 Nova Acta Acad. Petrop., 13, ad annos 1795-6, 1802 (1778), 45; Comm. Arith., II, 281. 



To conform with the notations of Euler's first paper, the interchange of a with p, b 

 with q, e with r, d with s has been made. Also, Opera postuma, 1, 1862, 246-9 (about 

 1777). 



