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
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-2}$ Fermat's Little Theorem: Exercise $5$