Prime iff Equal to Product
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Let $p \in \Z$ be an integer such that $p \ne 0$ and $p \ne \pm 1$.
Then $p$ is prime if and only if:
- $\forall a, b \in \Z: p = ab \implies p = \pm a \lor p = \pm b$
Let $p$ be a prime number.
Then by definition, the only divisors of $p$ are $\pm 1$ and $\pm p$.
Thus, if $p = a b$ then either $a = \pm 1$ and $b = \pm p$ or $a = \pm p$ and $b = \pm 1$.
- $\forall a, b \in \Z: p = a b \implies p = \pm a \lor p = \pm b$
This means that the only divisors of $p$ are $\pm 1$ and $\pm p$.
That is, that $p$ is a prime number.
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.1$. Arithmetic: Theorem $3$