MathNet 2018-part-05 — 181 problems (EN–VI) Question 88

A natural number $n$, $n\gt1$, is given. An integer $x$ is called splendid if the remainder of $x^{2}$ upon division by $n$ is odd. Prove that there are no more than $1+\lfloor\sqrt{3 n}\rfloor$ consecutive splendid natural numbers.