ON THE THEORY OF NUMBERS. 155 



gruence; it will be resoluble or irresoluble according as the index of A is or 

 is not divisible by d, the greatest common divisor of n andjo— 1, i.e. according 



as the exponent to which A appertains is or is not a divisor of ^H_ (see 



d 

 arts. 14 and 15). 



65. Solutio7i of Binomial Congruences. — We now come to the problem of 

 the actual solution of binomial congruences — a subject upon which our 

 knowledge is confined within very narrow limits. 



When a table of indices for the prime p has been constructed, the resolu- 

 tion of every binomial congruence, if it be resoluble, or, if not, the demon- 

 stration of its irresolubility, is implicitly contained in it. But to use a table 

 of indices for the solution of a binomial congruence is, as we have already 

 observed in a similar case (art. 16), to solve a problem by means of a recorded 

 solution of it. When the congruence x n =A, mod p, is resoluble, its solu- 

 tion may always be made to depend on that of a congruence of the form 

 x d ^a, mod p, where d is the greatest common divisor of n and p — 1, and 

 where a = A s , mod p, and ns=d, mod p— 1. We may therefore suppose 

 that, in the congruence x n =A, mod p, n is a divisor of p — 1. This con- 

 gruence (if resoluble at all) will have as many roots as it has dimensions ; if 

 i, be any one of them, and 1, 0,, 2 , . . . d»-i be the roots of the congruence 

 a*==l, modp, the roots of x n =A, modp, will be £, £0 l( £0 2 , ... £d n -i ; so that 

 the complete resolution of the congruence a.' a =A, mod p, requires, first, the 

 determination of a single root of that congruence itself, and, secondly, the com- 

 plete resolution of the congruence x"^^l, mod p. With regard to the first of 

 these requisites, in the important case in which the exponent t to which A 

 appertains is prime to n, a value ofx satisfying the congruence x n = A, mod p, 

 can be determined by a direct method (Disq. Arith. arts. 66, 67). For, in 

 this case, it will always happen that one value of a; is a certain power A k of 

 A, where k is determined by the congruence #» = 1, mod t. Nor is it 

 necessary, in order to determine k, to know the exponent t to which A 

 appertains; it is sufficient to have ascertained that it is prime to n; for, if 

 we resolve/*— 1 into two factors prime to one another, and such that one of 

 them is divisible by n and contains no prime not contained in n, the other 

 will be divisible by t, and may be employed as modulus instead of t in the 

 congruence %n=l, mod t. When this method is inapplicable, we can only 

 investigate a root of the congruence .t"=A, mod p (where A is different 

 from 1), by tentative processes, which, however, admit of certain abbreviations 

 (Disq. Arith. arts. 67, 68). The work of Poinsot (Reflexions sur la Theorie 

 des Nombres, cap. iv. p. 60) contains a very full and elegant exposition of 

 the theory of binomial congruences; but neither he nor anv other writer 

 subsequent to Gauss has been able to add any other direct method to that 

 which we have just mentioned. 



66. Solution of the Congruence x n =l, mod p. — When a single root of the 

 congruence x n =A is known, we may, as we have seen, complete its resolu- 

 tion by obtaining all the roots of the congruence £"=1, mod p. The methods 

 of Gauss, Lagrange, and Abel for the solution of the binomial equation 

 a-"— 1=0 are in a certain sense applicable to binomial congruences of this 

 special form. It is evident, from a comparison of several passages in the Dis- 

 quisitiones Arithmetica? *, that Gauss himself contemplated this arithmetical 

 application of his theory of the division of the circle, and that he intended to 

 include it in the 8th section of his work, which, however, has never been 

 given to the world. In fact, the method of Abelf which comprehends that 



* See Disq. Arith. arts. 61, 73, and especially art. 335. 



t See Abel's memoir, " Sur une classe particuliere d'equations resolubles algebriquement," 



