238 



REPORT — 1859. 



i. e. they represent the terms of a complete system of residues prime to p. 

 If y* = a, mod jo, k, or any number congruous to k for the modulus p — 1, is 

 termed the index* of a for the primitive root or base y ; and this is expressed 

 symbolically by writing 



K^Inda, mod (p— I), orK= Ind v a, mod (jo— !)• 



The principal properties of these indices, which it is clear are a kind of 

 arithmetical logarithm, are as follows : — 



(1) Ind(AB) = IndA+IndB,mod(jo-l). 



(2) Ind (A 5 ) = s Ind A, mod (p—1). 



(3) Ind (^, mod p\ = Ind A -Ind B, mod (jo-1). 



[The symbol/— , mod jo ) is used to denote the value of a: deduced from the 



congruence Bx = A mod jo.] 



(4-) Indy A = Ind y y'. Indy A, mod (jo— I). 



(5) If A = B, mod jo, Ind A == Ind B, modjo— 1. 



In these congruences A and B represent numbers prime to jo, s any inte- 

 gral number, and y and y two different primitive roots. 



The great importance of these indices in arithmetical researches has in- 

 duced the Academy of Berlin to publish a volume containing tables of the 

 numbers corresponding to given indices, and of the indices corresponding 

 to given numbers for all primes less than 1000. This volume, the 'Canon 

 Arithmeticusf,' was edited by C. G. J. Jacobi, and contains, besides the 

 Tables, a preface explaining the methods which he adopted in their construc- 

 tion. The annexed specimen will serve to exemplify the arrangement of the 

 Tables : — 



p=29 



jo-l=2 2 «7. 



Numeri. Indices. 



M. Burckhardt, to whom arithmetic is indebted for an excellent Table of 

 the divisors of numbers from 1 to 3,036,000 %, has inserted in his work, and 

 apparently only to fill up a blank page at the end of the first million, a 

 table stating the number of figures in the decimal period of the fraction 



-, for every prime number jo less than 2500. It is evident that the number 



* The reader must be careful to distinguish between the index of a number and the ex- 

 ponent to which the number appertains. The exponent does not depend on the choice of 



the primitive root : for a given number it has but one value, a, which is such that ^~~ is 



a 

 the greatest common divisor of the index aud of p — 1. The index may have any one of $ («) 

 different values ; which of these it has, depends on the particular primitive root chosen. 



t Berlin, 1839. 



J Paris, 1814-1817. A Table containing the exponents to which 10 appertains, for every 

 prime less than 10,000, has since been given by M. Desmarest. (See p. 308 of his ' Theorie 

 aes Nombres.') 





