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==Special Notation== | ==Special Notation== | ||
Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo. | Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo. | ||
+ | |||
+ | <math>(a_1, a_2,...a_n)</math> refers to the greatest common factor of <math>a_1, a_2, ...a_n</math>. | ||
+ | |||
==Properties== | ==Properties== | ||
For any number there will be only one congruent number modulo <math>m</math> between <math>0</math> and <math>m-1</math>. | For any number there will be only one congruent number modulo <math>m</math> between <math>0</math> and <math>m-1</math>. |
Revision as of 20:04, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 6. |
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
.