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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: |
2007-12-07 03:17:52 |
| Classification: |
msc:11A05 |
| Keywords: |
number theory |
| Defines: |
prime divisor, prime factor |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Each positive integer $n$ has a unique \PMlinkescapetext{decomposition} as a product
\[
n = \prod_{i=0}^{\ell} {p_i}^{a_i}
\]
of positive powers of its distinct positive {\it prime divisors} $p_i$. The {\it prime divisor} of $n$ means a (rational) prime number \PMlinkname{dividing}{Divisibility} $n$. A synonymous name is {\it prime factor}.
The decomposition is unique up to the \PMlinkescapetext{order} of the prime divisors and for\, $n=1$\, is an empty product.
For some results it is useful to assume that
$p_i < p_j$ whenever $i < j$. |
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