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Difference between revisions of "User:Temperal/The Problem Solver's Resource6"

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This section covers [[number theory]], especially [[modulo]]s (moduli?).
 
This section covers [[number theory]], especially [[modulo]]s (moduli?).
 
==Definitions==
 
==Definitions==
−
*<math>n\equiv a\pmod{b}</math> if <math>n</math> is the remainder when <math>a</math> is divided by <math>b</math> to give an integral amount.
+
*<math>n\equiv a\pmod{b}</math> if <math>n</math> is the remainder when <math>a</math> is divided by <math>b</math> to give an integral amount. Also, this means b divides (n-a).
 
*<math>a|b</math> (or <math>a</math> divides <math>b</math>) if <math>b=ka</math> for some [[integer]] <math>k</math>.
 
*<math>a|b</math> (or <math>a</math> divides <math>b</math>) if <math>b=ka</math> for some [[integer]] <math>k</math>.
 +
 
==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.

Revision as of 19:05, 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

  • $n\equiv a\pmod{b}$ if $n$ is the remainder when $a$ is divided by $b$ to give an integral amount. Also, this means b divides (n-a).
  • $a|b$ (or $a$ divides $b$) if $b=ka$ for some integer $k$.

Special Notation

Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo.

$(a_1, a_2,...a_n)$ refers to the greatest common factor of $a_1, a_2, ...a_n$ and $[a_1, a_2, ...a_n]$ refers to the lowest common multiple of $a_1, a_2,...a_n$.

Properties

For any number there will be only one congruent number modulo $m$ between $0$ and $m-1$.

If $a\equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $(a+c) \equiv (b+d) \pmod {m}$.

  • $a \pmod{m} + b \pmod{m} \equiv (a + b) \pmod{m}$
  • $a \pmod{m} - b \pmod{m} \equiv (a - b) \pmod{m}$
  • $a \pmod{m} \cdot b \pmod{m} \equiv (a \cdot b) \pmod{m}$

Fermat's Little Theorem

For a prime $p$ and a number $a$ such that $(a,b)=1$, $a^{p-1}\equiv 1 \pmod{p}$.

Wilson's Theorem

For a prime $p$, $(p-1)! \equiv -1 \pmod p$.

Fermat-Euler Identitity

If $gcd(a,m)=1$, then $a^{\phi{m}}\equiv1\pmod{m}$, where $\phi{m}$ is the number of relatively prime numbers lower than $m$.

Gauss's Theorem

If $a|bc$ and $(a,b) = 1$, then $a|c$.


Errata

All quadratic residues are $0$ or $1\pmod{4}$and $0$, $1$, or $4$ $\pmod{8}$.

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