Question

A cute problem on quadratic congruences

Answer 1

Let \(n\) be the number of integers in the set
\[\left\{ a, ~2a,~3a,~\ldots,~ \left(\frac{p-1}{2}\right)a\right\}\]
whose remainders exceed \(p/2\) when divided by \(p\).

Then prove that quadratic congruence \(x^2\equiv a \pmod{p}\) is solvable if \(n\) is even and not solvable if \(n\) is odd.

Answer 2

* \(p\) is an odd prime and \(a\) is any integer such that \(\gcd(a,p)=1\)

Answer 3

let me know if the problem statement is not clear.. il provide an example...