One is not Prime/Proof 2

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): $1$
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