IrMO 2017 Problem 6

Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?

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.