426 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



S. Roberts 125 treated the solution of x~2Py* = z 2 or 22 2 , when each 

 prime factor of P is of the form 8m+l. If P have one of the forms 



(8al) 2 + 16(2/3+ 1) 2 , (8fc3) 2 +8(2Z+l) 2 , (8fc-l) 2 -8(2Z) 2 , 



(8/c-3) 2 -8(2Z+l) 2 , 

 the equations 



2Py 2 = (8w3) 2 + (Sy3) 2 , 2Py 2 = 16^ 2 +2(8v3) 2 , 2Py 2 = 4u 2 - 2(8v I) 2 , 



are solvable. If, moreover, P is a prime of one of those forms or an odd 

 power of it, x 2 2Py 2 = 2 is solvable. There are three more such triples of 

 equations leading to analogous conclusions. 



T. Pepin 126 proved Legendre's criterion as quoted by Gauss. 116 

 G. Heppel 127 treated d 2 = 2a 2 +b 2 by setting b = d-2f, whence 



Thus a is even, a = 2q. Hence express 2g 2 as a product fh and take d = h-\-f, 

 b hf, a = 2q. 



F. Goldscheider 128 expressed in terms of one solution the general solution 

 of ax 2 +fa/ 2 +C2 2 = which satisfies the final congruences of Dirichlet. 119 

 He proved that there exists such a solution for which also kx-\-k'y-\-k"z 

 is relatively prime to a given odd integer s, if k } k', k" are given integers 

 whose g.c.d. is prime to s. 



G. de Longchamps 129 wrote x 2 = y 2 -\-pz 2 in the form 



(x+y}/(pz)=z/(x-y}=t. 



Hence t must divide z. Set z = 2\t. Thus x = \(pt 2 -\-l), y = \(pt 2 !), 

 where X and t are arbitrary. For nx 2 = y 2 +(n l)z 2 , see de Longchamps. 162 



P. Bachmann 130 gave a clear exposition of our subject. 



R. P. Paranjpye 131 proved that all integral solutions of x 2 z 2 2y 2 are 



where X, ju are relatively prime. Since y is even, 



A. S. Werebrusow 132 noted that, if a 2 Dj3 2 = 7na 2 , a second set of solutions 

 of X 2 DY 2 = mZ 2 is given by 



' = o- ac. 



A. Cunningham, 133 to solve x z +y z = Az 2 , used the known solutions 

 Y=(P+Au 2 }d, Z = 2tufd, x = (P-Au*)/d, of Y 2 -AZ 2 = x 2 , where d = l or 2, 

 and solutions of r z -Av 2 =-l. Then (F 2 -AZ 2 )(r 2 -Av 2 ) = -z 2 , whence 



give the general solutions. A. Holm (p. 70) 



Proc. London Math. Soc., 11, 1879-80, 83-87. 



126 Atti Accad. Pont, Nuovi Lincei, 32, 1878-9, 88. 



127 Math. Quest. Educ. Times, 38, 1883, 56. 



128 Das Reziprozitatsgesetz der achten Potenzreste, Progr., Berlin, 1889, 8. 



129 El Progreso Mat., 4, 1894, 46; Jour, de math. 616m., 18, 1894, 5. 



130 Arith. der Quad. Formen, 1898, 198-224, 231. 



131 Math. Quest. Educ. Times, 75, 1901, 119. Cf. papers 109, 116 of Ch. XVI. 



132 Mem. Sc. Univ. Moscow, 23; 1'intermc'diaire des math., 9, 1902, 187. 



133 Math. Quest. Educ. Times, (2), 9, 1906, 69-70. 



