Euler’s Criterion for Quadratic Residues

Leonhard Euler (source: Prolineserver)

Here we would show a beautiful application of Fermat’s Little Theorem.

An integer a is a quadratic residue with respect to prime p if a \equiv x^2 \mod{p} for some integer x.

Given a prime p and an integer a, how fast can we decide whether a is a quadratic residue modulo p? As it turns out (courtesy to Euler), pretty fast.

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Fermat’s Little Theorem


This is a beautiful theorem which states that the difference between a positive integer a and a prime power of a, a^p-a, is divisible by p. It’s history can be found in Wikipedia: Fermat posited the theorem (and as usual, did not give the proof) while Euler first published a proof using mathematical induction. Below, we will state the theorem and provide a simple-to-understand proof using only modular arithmetic, followed by another simple proof using mathematical induction.

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