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 $n.$

Corollaries: 1
Definitions: 2 3 4 5
Explanations: 6
Proofs: 7 8 9 10 11 12 13 14 15 16 17 18
Propositions: 19 20 21 22 23
Sections: 24 25
Theorems: 26

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition