# One is not Prime

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## Theorem

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.

$\blacksquare$

## Proof 2

From Sigma Function of Prime Number, the sum $\map \sigma p$ of all the positive integer divisors of a prime number $p$ is $p + 1$.

But from Sigma Function of 1, $\map \sigma 1 = 1$.

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

$\blacksquare$

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-2}$ Divisibility - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Primes

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