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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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