MK-MO 2017 Problem 2

Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square.

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.