ON THE THEORY OP NUMBERS. 239 



of terms in the decimal period of - is nothing else than the exponent to 



P 

 which 10 appertains for the modulus p. M. Burckhardt's table, therefore, 

 at once apprises us that out of the 365 primes inferior to 2500 (2 and 5 are 

 not counted in this enumeration, as being divisors of 10), 10 is a primitive 

 root of 148 ; because there are 148 primes p below 2500, the reciprocals of 

 which have decimal periods consisting of p— 1 figures. Again, for 108 of 

 the remaining primes below 2500, the exponent to which 10 appertains is 

 \{p — \). Of these 108 primes, 73 are of the form 4w + 3, from which it 

 maybe inferred that —10 is a primitive root of those 73 numbers. M. 

 Burckhardt's Table supplies us, therefore, with a primitive root (and that 

 root the most convenient for the purposes of computation) of 148 + 73=221 

 out of the 365 primes inferior to 2500. Nor is this the limit to its useful- 

 ness ; for when the exponent to which 10 appertains is as high as ^ (p— 1) 

 or i (p — 1) or i (p— 1), it is possible by methods which Jacobi has indicated 

 to construct the Table of Indices with very little labour. 



Jacobi says that had it not been for this table of Burckhardt's he should 

 hardly have ventured on the construction of the ' Canon Arithmeticus,' on 

 account of the prolixity and uncertainty of the tentative methods for the in- 

 vestigation of primitive roots. But, while endeavouring to avail himself of 

 the results of M. Burckhardt's table, for the computation of his own Tables 

 of Indices, in other cases besides those in which that Table immediately fur- 

 nishes a primitive root, he was led to the invention of a general method of 

 procedure, which, as he says, would have enabled him to dispense with the 

 assistance of Burckhardt's Table altogether, or to extend his Canon to any 

 higher limit which the expense of printing would have admitted. This 

 method is not in principle very different from Gauss's process for finding 

 primitive roots, but the form which Jacobi has given to it possesses great 

 advantages, for the purpose to which he has applied it. He first of all takes 

 a number a (not quite at hap-hazard, for quadratic residues can at any rate 

 be excluded by the law of reciprocity ; see inf. Art. 16) ; and determines its 

 period of residues, and the exponent a to which it appertains. Let aa'=p — 1, 

 and let the residues of a, a 2 , a 3 . . . a a , be entered in a Table of which the argu- 

 ments are the indices 1, 2,3, ...p — 1, opposite to the indices, a', 2a',3a'. . . aa, 



respectively. It has been shown by Gauss that there are always ^ ^ ' 



primitive roots for which this assignment is true. A number o is then taken, 

 not contained in the period of a, and the residues of its successive powers are 

 formed till we come to the lowest power of it that is congruous to any power 

 of a ; so that b B = a A , mod p. Let j3 be the exponent to which b appertains, 



the greatest common divisor of a and /3, and \=-^- their least common 







multiple; let also /3/3'=jo— ]. It may be proved thatB=— ; A = — ; where 



6 6 



k is some number less than G and prime to it, so that - is the greatest com- 

 mon divisor of A and a. These relations show, that when we know the 

 numbers a. A, and B, we can immediately find d, k, and /3, without having 

 to raise b to any power higher than b B . We may then assign to b any index 

 of the form //3', where I is prime to /S, and congruous to k for the modulus 0. 



The number of such values of / (incongruous for the modulus /3) is ^ , ( l 



