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'fundamental theorem of arithmetic'
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| Title of object: |
fundamental theorem of arithmetic |
| Canonical Name: |
FundamentalTheoremOfArithmetic |
| Type: |
Theorem |
| Created on: |
2001-10-15 20:50:09 |
| Modified on: |
2005-09-19 19:35:41 |
| Classification: |
msc:11A05 |
| Keywords: |
number theory |
Revision comment (for changes between this and next version):
| Changes for correction #7097 ('"Defines also"'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Each natural number $n \geq 1$ can be decomposed uniquely, up to the order of the factors, as a product of positive prime numbers. This allows us to write $n$ in the unique representation
$$
n = {p_1}^{a_1} {p_2}^{a_2} {p_3}^{a_3} \cdots {p_k}^{a_k}
$$
for some nonnegative integer $k$ with $a_l$ positive integers, $p_i$ positive primes and $p_i \neq p_j$ for $i \neq j$. For some results it is also useful to assume that $p_i < p_j$ for $i < j$. The $p_i$ are called the prime factors of $n$. |
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