Remainder of an integer
modulo
, residue of
modulo
Any integer which is congruent to
modulo
(cf. Congruence). Let
be the remainder of division of
by some integer
,
; then the residue
of the number
modulo
will have the form
, where
is some integer. The residue corresponding to
is equal to
and is called the least non-negative residue of
. The smallest (in absolute value) residue
is called the absolutely smallest residue of
. If
, then
; if
, then
; finally, if
is even and
, either
or
may be taken as
.
A system consisting of integers each one of which is the residue of one and only one of the numbers
is called a complete system of residues modulo
. The smallest non-negative residues
or the absolutely smallest residues are the complete systems of residues which are most frequently used.
A power residue of degree modulo
,
, is any integer
, coprime with
, for which the congruence
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is solvable. If this congruence is not solvable, is called a power non-residue of degree
modulo
. In particular, if
, the residues or non-residues are called quadratic; if
, they are called cubic; if
, they are called biquadratic (see also Power residue).
References
[1] | I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) |
Comments
References
[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII |
Remainder of an integer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Remainder_of_an_integer&oldid=13893