ON THE THEORY OF NUMBERS. 233 



(2, 1 ) a?, + (2, 2) a? a + (2, 3) * 3 + . . + (2, n) *„=«, (A). 



(«, 1) .r 1 + (w, 2) a? a +(w, 3) x 3 + . . + (», ») a? B ==a„; 



let the modulus be q, and the determinant 2+ (1, 1) (2, 2) . . (», w)= D « 

 If the determinant be prime to the modulus, these congruences will always 

 admit of one, and only one, system of solutions, namely, that supplied by the 

 system of congruences 



~ k = n dD 



Dx ' = S TFT—- \ Uk ' 

 h=\ d{k,r) 



But if D be not prime to q, let q=p x mi .p.™* where p v p# &e. denote 



different primes. In order that the proposed system should be resoluble for 

 the modulus q, it must be separately resoluble for each of the modules p™ 1 , 

 p™ 2 , &c. ; and conversely if it be resoluble for each of those modules, and 

 admit P, solutions when taken with respect to the modulus p™ 1 , P 2 solutions 

 when taken with respect to the modulus jo 2 '" 3 , and so on, it will be also 

 resoluble for the modulus q, and will admit P : X P„ X P 3 . . .solutions for that 

 modulus. It is, therefore, only necessary to assign the number of solutions 

 of the congruences (A), for a modulus p m which is the power of a prime. 

 Let I be the index of the highest power of p which divides D ; and similarly 

 let I denote the index of the highest power of p which divides all the 

 minors of D which are of order r ; then if I„— I„_i < m, the system (A) (if 

 resoluble at all) admits of p ln solutions; but if I„ >»i + I«_i, it will always 

 be possible, in the series of differences 



're — Ire— 1> *n— 1 — *n— 2> •■ • ■ 



to assign a pair of consecutive terms I r +i — 1» I,- — 1»— 1» satisfying the in- 

 equalities 



I, + i— I r >m>I r — l r -\ ; 

 and then the number of solutions (supposing always that the congruences 

 are resoluble) is expressed by the formula pir+(n-r)m m 



The analogy of this theory with the corresponding algebraic theory of 

 systems of linear equations is in particular cases very striking. For example, 

 we have in Algebra the theorem 

 "The system of n linear equations 



(1, 1) x x + (1, 2) ar 2 + (1, 3) x a + . . . + (1, ») x n =0 

 (2, 1) *,+ (2, 2) x 2 + (2, 3) x, + ... + (2, n) x n =0 



(n, 1) x x + (», 2) x t + (n, S)x a + ... + (n, w) x n =0 

 implies either that D=2+ (1,1) (2, 2) . . . («, ?i)=0, or else that x x x 2 . . . x n 

 are separately equal to zero." 



In the Theory of Numbers we have the corresponding theorem, 

 " If n linear and homogeneous functions of an equal number of indetermi- 

 natcs be congruous to zero for a prime modulus, either the determinant of 

 the system is congruous to zero for that modulus, or else every one of the 

 indeterminates is separately congruous to zero." 



10. JFermat's Theorem. — The theory of congruences of the higher orders 

 is so essentially connected with Fermat's Theorem, that it will be proper 

 before proceeding further to introduce a few considerations relating to that 

 celebrated proposition. 



