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The fundamental theorem of arithmetic motivates the following definition:

Definition: Canonical Representation of Natural Numbers, Factorization

Given consecutive prime numbers \(p_1=2, p_2=3, p_3=5, p_4=7, p_5=11,\ldots\) we can write each natural number \(n \ge 1\) as a product
\[n=\prod_{i=1}^\infty p_i^{e_i}.\]
According to the above theorem, the product is unique for each \(n > 1\) and we call it the canonical representation of \(n\). By setting the canonical representation of \(1\) to
\[1=\prod_{i=1}^\infty p_i^0,\]
we can extend the definition to \(n \ge 1\). Please note that for each \(n \ge 1\) its canonical representation is actually a finite product, since only finitely many exponents \(e_i\) are different from \(0\).

Sometimes, it is more convenient to choose indexing of primes, which depends on the number $n$ is such a way that $p_1,\ldots,p_r$ are exactly those primes, which divide $n.$ In this case the product \[n=\prod_{i=1}^r p_i^{e_i}\]
the factorization of $n.$

| | | | | created: 2014-08-24 09:33:50 | modified: 2019-04-07 07:51:28 | by: bookofproofs | references: [701]

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Bibliography (further reading)

[701] Scheid Harald: “Zahlentheorie”, Spektrum Akademischer Verlag, 2003, 3. Auflage

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