What is the most motivating way to introduce quadratic residues? I would like some concrete examples which have an impact. This is for first-year undergraduates doing an elementary number theory course. They have done Diophantine equations, solved linear congruences, primitive roots.
2026-02-22 19:30:31.1771788631
How to introduce quadratic residues?
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I like to build the quadratic part of elementary number theory on the Diophantine equation $$ x^2 + dy^2 = n. $$ When $n$ is prime a necessary condition is that $-d$ is a quadratic residue.
You can start this study with the case $d = 1$. When does $-1$ have a square root modulo a prime $p$? That's approachable without too much machinery. Lots of good history.