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



J. Plana 120 stated that all integral solutions of x 1 79i/ 2 = lOlz 2 are given by 



x = a 927p 2 +- 4572# 2 +3126pg, 

 a 



i 414g 2 + 



30pq, 



a 



for a = 2 or 1, where p, q are arbitrary integers. 



G. Cantor 121 considered the solution in integers of F = 0, where F is 

 any ternary quadratic form. A formal solution (<, \f/, x) is one for which 

 F(0, \p, x) = identically in x, y, where 0, are binary quadratic forms in 

 x, y. In particular, let F be 



Let the greatest common divisor of the three coefficients of 4>, and those for 

 \f/ and x be relatively prime in pairs; then the formal solution (<, \l/, x) 

 is primitive, and we can find integers w's for which 



a'\[/ (mod a) 

 a"x (mod a') 

 w"4>=a'\}/, w"\}/=a4> (mod a"), 



identically in x, y. By the two congruences in the first line, 



(w> 2 +a'a")fo=0 (mod a). 



Then t0 2 +a'a" = if a is odd, or when a is even if \f/, x are properly primitive. 

 The solution (<, \f/, x) is said to pertain to the combination {w, w', w"} if 



w z +a'a" = (mod a), w' 2 +aa" = (mod a'), w" 2 +aa' = Q (mod a"). 



The number of possible sets of roots is 2 a)+7? , where w is the number of distinct 

 odd prime factors of the determinant D=aa'a" of the primary form 

 [aaV], while 77 = 0, 1 or 2, according as D/4 is not integral, an odd or even 

 integer. Then, if a' a", a" a, aa' are quadratic residues of a, a', a", 

 respectively, there is a primitive solution (<, ^, x) of [aa'a"] = pertaining 

 to any chosen one of the 2"+^ combinations {w, w', w"}, and [aa'a // ] = 

 has exactly cr-2 ia+ri systems of primitive solutions, where a = 2 if D=0 

 (mod 4), while a = 4 in all other cases. 



L. Calzolari 122 treated (7) u 2 = Ax-+Bf by setting (8) u=Yx+Xy. 

 The discriminant of the resulting quadratic in x, y is to be a square, whence 



(9) AX 2 +BY 2 -AB=U 2 . 



Eliminate X between the latter and (8), using (7). We get U z y z = (uY Ax} z , 



(10) Ax=YuUy, By = Xu=FUx. 



Thus from a set of solutions of (9), we get one of (7), viz., that given by (8) 



120 Memorie R. Accad. Torino, (2), 20, 1863, 107, footnote. 



121 De Aequat. secundi Gradus indet., Diss. Berlin, 1867. 



122 Giornale di Mat., 7, 1869, 177-192. 



