BG-JBMO-TST 2018 Problem 3

Find all positive integers $n$ such that the number $n^6 + 5n^3 + 4n + 116$ is the product of two or more consecutive numbers.

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.