One is not Prime/Proof 2
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Theorem
The integer $1$ (one) is not a prime number.
Proof
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 $\map {\sigma_1} p = p + 1$.
$\blacksquare$
Historical Note
The reasoning that $1$ should be excluded from the set of prime numbers based on its divisor sum was by Leonhard Paul Euler.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1$