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The proof and various concrete examples make the lemma and application clear.

I was wondering if there is an intuitive (or other) perspective so that if you knew the number of negative least residues $\pmod p$ of $a$, $2a$,...$((p - 1)/2)a$, you would know whether $a$ is a quadratic or non-quadratic residue $\pmod p$. Where $a\in \mathbb{Z}$, $p$ an odd prime, and $p$ does not divide $a$.

Thanks

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