366 report — 1865. 



" Let {a, h, c) represent in succession every uneven reduced form of the de- 

 terminants — p, — p + l\ —p+2\ ; only, if a=c, let the reduced form 



satisfy the special condition (art. 92) 6<0, instead of 6>0 ; the roots of 

 the congruences 



a w 2 + 2bu> + c == 0, mod p, 



of which roots the number is 



r(p)+2F(2.-l 2 ) + 2F( P -2 2 ) + 



are a complete system of residues for the modulus p." 



As it appears from the formula V. that the number of these congruence- 

 roots is equal to jp, it is only necessary to prove that they are all different ; 

 the demonstration of this very difficult point M. Kronecker has effected by 

 showing that the contrary supposition is inconsistent with the inequalities 

 satisfied by the coefficients of the reduced forms. A proof, independent of the 

 formula V., that every residue of p is a root of one of the congruences, would 

 of course supply a direct arithmetical proof of that formula, for the case in 

 which n is a prime of the form Am + 3. 



Arithmetical demonstrations of the formulae of M. Kronecker have also 

 been obtained by M. Liouville. These demonstrations depend on the prin- 

 ciples introduced by him into arithmetic in the series of memoirs " Sur quel- 

 quesformules generates qui peuvent etre utiles dans latheorie des nombres"*, 

 and originally suggested (as he himself informs us) by Jacobi's arithmetical 

 proof of the theorem of four squares. M. Liouville has given, as an example 

 of his method, a proof of the equation (XI.)-(XIL), or 



4F(n-l 2 ) + 4F(«-3 2 )+4F(»i-5*) + . . =A(n)-l\n)-T'(n), 



for the two cases in which n is unevenly, and evenly, event. "We shall con- 

 fine our attention to the latter and somewhat simpler case. It requires two 

 preliminary Lemmas, both included as very particidar cases in M. Liouville's 

 general formulae. 



I. Let m represent a given uneven number, a. a given positive exponent 

 other than zevo,f(x) an even function, so that /(#)=/( — x) ; we have the 

 equation 



2[f(d' -d") -f{d' + d")] = 2*-iZd[f(0)-f(2*d )], 



the summations in the left and right-hand members extending respectively to 

 all solutions of the equations 



2*m=d'c' + d"c",m = dt, 



the indeterminates d', d", §', 8" in the first equation, and d, I in the second, 

 being positive and uneven, and two solutions of either equation being re- 

 garded as different, unless the indeterminates of the two solutions are the 

 same and in the same order. 



To establish this equation, we consider the system 



d'V + d"h"=2 a m^ 

 d' + d" =2fi (a) 



r— a" =2 V J 



in which \i and v are given positive integers. The solutions of this system 

 are equal in number to the solutions of the system 



d'V+d"h" = 2 a m "I 

 d'—d" =-2p (a') 



a'+a" =2v J 



* Liouville's Journal, rols. iii.-viii. (New Series.) 

 t Liouville, New Series, vol. vii. p. 44. 



