Prime iff Coprime to all Smaller Positive Integers
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Theorem
Let $p$ be a prime number.
Then:
- $\forall x \in \Z, 0 < x < p: x \perp p$
That is, $p$ is relatively prime to all smaller (strictly) positive integers.
Proof
From Prime not Divisor implies Coprime, if $p$ does not divide an integer $x$, it is relatively prime to $x$.
From Absolute Value of Integer is not less than Divisors: Corollary, $p$ does not divide an integer smaller than $p$.
It follows that $p$ is relatively prime to all smaller (strictly) positive integers.
The special case when $x = 0$ is excluded as from Integers Coprime to Zero, $p$ is not relatively prime to $0$.
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$