I'm working on the following proof

Let p be an odd prime number. Prove that the $\frac{p-1}{2}$ quadratic residues modulo p are congruent to $$1^2,2^2,3^2,...,\left(\frac{p-1}{2}\right)^2$$

I'm not really sure how to start this problem. What should I be looking to do?


1 Answer 1



are quadratic residues because they have trivial solutions : $$1,2,3,\ldots, \dfrac{p-1}{2}$$

Next notice that $a^2\equiv (p-a)^2 \pmod{p}$ to conclude that the given list exhausts the incongruent quadratic residues modulo $p$.


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