Square Modulo 3/Corollary 3

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Corollary to Square Modulo 3

Let $n \in \Z$ be an integer such that:

$3 \nmid n$

where $\nmid$ denotes non-divisibility.

Then:

$3 \divides n^2 - 1$

where $\divides$ denotes divisibility.


Proof

From Square Modulo 3:

$n \equiv 0 \pmod 3 \iff n^2 \equiv 0 \pmod 3$

Hence also from Square Modulo 3:

$n \not \equiv 0 \pmod 3 \iff n^2 \equiv 1 \pmod 3$


That is: $3 \nmid n \iff 3 \divides n^2 - 1$

$\blacksquare$


Sources