MathNet 2019-part-07 — 184 problems (EN–VI) Question 30

Let $p = 8k + 1$ be a prime, where $k$ is a positive integer. Let $r$ be the remainder when $\binom{4k}{k}$ is divided by $p$ (so $0 \le r \lt p$). Prove that $\sqrt{r}$ is not an integer.