of a number 
modulo 
The exponent 
in the congruence 
, where 
and 
are relatively prime integers and 
is a fixed primitive root modulo 
. The index of 
modulo 
is denoted by 
, or 
for short. Primitive roots exist only for moduli of the form 
, where 
is a prime number; consequently, the notion of an index is only defined for these moduli.
If 
is a primitive root modulo 
and 
runs through the values 
, where 
is the Euler function, then 
runs through a reduced system of residues modulo 
. Consequently, for each number relatively prime with 
there exist a unique index 
for which 
. Any other index 
of 
satisfies the congruence 
. Therefore, the indices of 
form a residue class modulo 
.
The notion of an index is analogous to that of a logarithm of a number, and the index satisfies a number of properties of the logarithm, namely:
where 
denotes the root of the equation
If 
is the canonical factorization of an arbitrary natural number 
and 
are primitive roots modulo 
, respectively, then for each 
relatively primitive with 
there exist integers 
for which
The above system 
is called a system of indices of 
modulo 
. To each number 
relatively prime with 
corresponds a unique system of indices 
for which
where 
, 
, and 
and 
and defined as follows:
Every other system 
of indices of 
satisfies the congruences
The notion of a system of indices of 
modulo 
is convenient for the explicit construction of characters of the multiplicative group of reduced residue classes modulo 
.
References
| [1] | I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) |
References
| [a1] | H. Davenport, "Multiplicative number theory" , Springer (1980) |
How to Cite This Entry:
Index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Index&oldid=47331
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
