474 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



etc., may become unity. Then, if a = s = l, we have N=(p 2 +l)(r z +q y )[q z . 

 If p = 7, g = 5, 2 divides p 2 +l, and N = 2r 2 +50. If a= -1, s = l, then 

 J\r=(p 2 -l)(r 2 -g 2 )/g 2 and ^ divides p 2 -! if p = 3, = 2, etc. A list is 

 given of the resulting N's numerically <100; those ^50 and >0are7, 10, 

 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50. But the problem is 

 not proved impossible for the omitted values of N. 



Euler 81 made a 2 x 2 +b 2 y 2 and a 2 y 2 -\-b 2 x 2 both squares by taking 



ax_p 2 q 2 ay_r 2 s 2 



by 2pq bx 2rs 



By division, we get x 2 /y 2 . Hence it suffices to make the quotient of 

 pq(p 2 q 2 } by rs(r 2 s 2 ) a square, a problem* which had been frequently 

 treated, but not completely solved. The first of three special methods is 

 to take s = q, r = p-\-q; then we are to make (p q}/(p+2q} = D, which is 

 the case if p = u 2 +2t 2 , q = u 2 t 2 ; the resulting solution is 



a = 3tu, b = 2(u 2 -f), x = t(2u 2 +t 2 ), y = u(u 2 +2t 2 ). 



To obtain the general solution, we may take s = q without loss of generality, 

 since it is only a question of ratios. Then 



_p(p 2 -q 2 )_ p 3 -nr 3 



' / o o\ i i) q 



r(r 2 q 2 ) p nr 



Set p = rv. Then (v* ri)l(vri) = D = (vz) z if 



(n+2z)v 2 -z(2n+z)v+n(z z -l)=Q. 

 From a given solution z, v, we get a second solution 



, z(z+2n) , , 



v=- - v, z =2vz. 



Thus v = 0, z = l leads to the second solution 



l+2n Zn 



' z' = 



2+n' 2+n 



Replace n by t 2 ju 2 ; we get the above special solution. He investigated the 

 third solution v", z", and also started with y = 0, 2=!; z = 0, y=l; 

 y=oo. Further, he treated the general condition for n = 4 and n = l/4. 

 In conclusion, he found a, , d such that 



a 2 b 2 +c 2 d 2 , a 2 c 2 +b 2 d 2 , a 2 d 2 +b 2 c 2 

 are all squares. For f=t 2 3u 2 , g = 2tu, we have 



Then a solution is a = 2g, b = 2h, c=f+g, d=fg, and the three quartics 

 are the squares of / 2 +7gr 2 , 2(/ 2 =F/gr+20 2 ). 



C. F. Degen 82 treated x 2 -\-my 2 = p 2 , x 2 +ny 2 = q 2 . We may set 



p = a(mt-\-nu), q = o.(nt-\-mu) . 



81 Mem. Acad. Sc. St. P6tersbourg, 11, 1830 (1780), 12; Comm. Arith., II, 425-37. 



* Euler, 67 seq., and Euler" of Ch. IV, Petrus 12 and Euler 33 of Ch. XV, Euler 18 - 19 of Ch. 

 XVIII, Euler 253 of Ch. XXII. 



82 M6m. prdsente"s acad. sc. St. P6tersbourg par divers eavans, 1, 1831 (1823), 29-38. 



