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Number Theory
This section covers number theory, especially modulos (moduli?).
Definitions
if
is the remainder when
is divided by
to give an integral amount.
(or
divides
) if
for some integer
.
Special Notation
Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo.
refers to the greatest common factor of
.
Properties
For any number there will be only one congruent number modulo between
and
.
If and
, then
.
Fermat's Little Theorem
For a prime and a number
such that
,
.
Wilson's Theorem
For a prime ,
.
Fermat-Euler Identitity
If , then
, where
is the number of relatively prime numbers lower than
.
Gauss's Theorem
If and
, then
.
Errata
All quadratic residues are or
and
,
, or
.