Square Modulo n Congruent to Square of Inverse Modulo n

Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:

$a^2 \equiv \paren {n - a}^2 \pmod n$

where the notation denotes congruence modulo $n$.

Proof

\(\ds \paren {n - a}^2\)

\(=\)

\(\ds n^2 - 2 n - a^2\)

\(\ds \)

\(\equiv\)

\(\ds a^2\)

\(\ds \pmod n\)

$\blacksquare$