Erdős Problem 1065

Reference: erdosproblems.com/1065

theorem Erdos1065.erdos_1065a : {p : ℕ | ∃ (q : ℕ) (k : ℕ), Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * q + 1}.Infinite ↔ sorry

Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$?

This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. Guy

theorem Erdos1065.erdos_1065b : {p : ℕ | ∃ (q : ℕ) (k : ℕ) (l : ℕ), Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * 3 ^ l * q + 1}.Infinite ↔ sorry

Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$?