CHAP. XIX] z 2 +?/ 2 , z 2 -f-2 2 , if+z- ALL SQUARES. 501 



J. Neubcrg 17 satisfied Euler's 3 condition p 2 +g 2 4+l/p 2 -f-l/2 2 = w; 2 by 

 the special values w=2/(pq), p 2 +q 2 = 4. The latter holds if 



4rs _2(r 2 -s 2 ) 



~r 2 +s 2 ' q= r 2 +s 2 ' 

 Hence x 2 +?/ 2 =f 2 , ?/ 2 +z 2 = 2 , 2 2 +z 2 = T? 2 for 



z = 8rs(r'-s 4 ); = (r 2 +s 2 ) 3 ; y, r = (r 2 -s 2 ) {(V 2 +s 2 ) 2= Fl6rV> ; 



z, 77 = 2rs{(r 2 +s 2 ) 2 T4(r 2 -s 2 ) 2 }. 



C. Leudesdorf 18 solved the equivalent system 2(u 2 -\-v 2 w 2 )=x 2 , 

 2(u 2 +w 2 v 2 )=y 2 , 2(v z -\-w 2 u 2 }=z 2 by use of trigonometric functions (cf. 

 Gill 13 ). G. Heppel repeated Neuberg's 17 solution. 



J. Matteson 19 obtained Euler's 4 result by the method of Euler. 3 



A. Martin 20 varied Scott's 15 method by making the first two terms of the 

 quartic in x cancel (giving x = 2mn/s), instead of the last two. 



K. Schwering 21 proceeded as had Neuberg 17 with X, n in place of p, q. 

 To connect the result with elliptic functions, set R(p)=p*+l-\-p 2 p, 



Then p is a well-known elliptic function. By the addition theorem, 



Hence if $(u) and $'(u) are rational, also \f/(2u], ^(3w), , ^'(2u), 

 are rational. Thus one solution p, q yields an infinitude of solutions. The 

 relation of the same problem to Abel's theorem is considered on p. 11. 



Several writers 22 gave solutions. 



* F. Ferrari 23 gave an infinitude of solutions. 



R. F. Davis 24 gave Neuberg's 17 solution. 



A. Martin 24 " gave another derivation of Euler's 6 result. 



H. Olson 246 proved that, if x 2 +y 2 = u 2 , x 2 +z z = v*, y 2 +z 2 = w 2 , the product 

 xyzuvw is divisible by 3 4 -4 4 -5 2 . 



M. Rignaux 240 stated that all solutions of x 2 +i/ 2 = D, etc., are given by 



x = 2mnpq, y = mn(p 2 q 2 ), z = pq(m 2 n 2 }, 7/ 2 +z 2 =D, 

 and noted four solutions, involving parameters, of the final condition. 



17 Nouv. Corresp. Math., 1, 1874-5, 199-202. 



18 Math. Quest. Educ. Times, 34, 1881, 95-6. 



"Collection of Diophantine Problems . . ., ed., Martin, Washington, D. C., 1888, 21. 



20 Math. Magazine, 2, 1898, 214-5. 



21 Geom. Aufgaben mit rationalen Losungen, Progr., Diiren, 1898, 9. 



22 Amer. Math. Monthly, 6, 1899, 123-5; Math. Quest. Educ. Times, 68, 1898, 104; (2), 11, 



1907, 26-7. 



"Suppl. al Periodico di Mat., 14, 1910-11, 138-140. 



24 Math. Quests., and Solutions, 2, 1916, 24-25; Math. Mag., 2, 1898, 215. 

 Ma Amer. Math. Monthly, 25, 1918, 305-6. 

 246 Ibid., 304-5. 

 24c L'intermediaire des math., 25, 1918, 127. 



