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 .
|||I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)|
|[a1]||H. Davenport, "Multiplicative number theory" , Springer (1980)|
Index. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Index&oldid=17202