DE-MO 2006 Problem 1

Find all $n\in \mathbb Z^+$ , so that $z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} }$ is 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.