One is not Prime

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 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 $\map {\sigma_1} p = p + 1$.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility
• 1979: G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers (5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.2$ Prime numbers
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1$
• 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