One is not Prime

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The integer $1$ (one) is not a prime number.

Proof 1

By definition, a prime number is a positive integer which has exactly $2$ divisors which are themselves positive integers.

From Divisors of One, the only divisors of $1$ are $1$ and $-1$.

So the only divisor of $1$ which is a positive integer is $1$.

As $1$ has only one such divisor, it is not classified as a prime number.


Proof 2

From Divisor Sum of Prime Number, the sum $\map {\sigma_1} p$ of all the positive integer divisors of a prime number $p$ is $p + 1$.

But from Divisor Sum of 1, $\map {\sigma_1} 1 = 1$.

If $1$ were to be classified as prime, then $\map {\sigma_1} 1$ would be an exception to the rule that $\map {\sigma_1} p = p + 1$.