One is not Prime/Proof 1

Theorem

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


Proof

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.

$\blacksquare$


Sources

  • 1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers
  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.