= Prime-exponent parametrization of smooth numbers
{title2=$p_1^{e_1}\cdots p_r^{e_r}\longleftrightarrow(e_1,\ldots,e_r)$}
For distinct <primes> $p_1,\ldots,p_r$, let $M$ consist of the positive <integers> having no other <prime factors>, including $1$. The map from a number to its exponent tuple is a <bijection> from $M$ to $\mathbb N^r$ when $0\in\mathbb N$. The <fundamental theorem of arithmetic> proves that the tuple is uniquely defined and that equality of tuples implies equality of numbers. Every nonnegative tuple gives the allowed number $p_1^{e_1}\cdots p_r^{e_r}$, proving <surjectivity>. Multiplication of numbers corresponds to coordinatewise addition of tuples, with $1$ corresponding to the zero tuple.
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