IL-GILLIS 2021 Problem 2

Does there exist an infinite sequence of primes $p_1, p_2, p_3, \dots$ for which, $p_{n+1}=2p_n+1$ for each $n$ ?

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.