# Tau of Prime Number

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## Theorem

Let $p \in \Z_{> 0}$.

Then $p$ is a prime number if and only if:

- $\map \tau p = 2$

where $\map \tau p$ denotes the tau function of $p$.

## Proof

### Necessary Condition

Let $p$ be a prime number.

Then, by definition, the only positive divisors of $p$ are $1$ and $p$.

Hence by definition of the tau function:

- $\map \tau p = 2$

$\Box$

### Sufficient Condition

Suppose $\map \tau p = 2$.

Then by One Divides all Integers we have:

- $1 \divides p$

Also, by Integer Divides Itself we have:

- $p \divides p$

So if $p > 1$ it follows that $\map \tau p \ge 2$.

Now for $\map \tau p = 2$ it must follow that the only divisors of $p$ are $1$ and $p$.

That is, that $p$ is a prime number.

$\blacksquare$