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=17202
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article