CAPS 2008 Problem 3

Find all primes $p$ such that the expression $\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2$ is divisible by $p^3$ .

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.