FormalConjectures.ErdosProblems.«1065»

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import FormalConjecturesUtil

Erdős Problem 1065

Reference: erdosproblems.com/1065


namespace Erdos1065

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

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1065.parts.i :
    answer(sorry) ↔ Set.Infinite {p | ∃ q k, p.Prime ∧ q.Prime ∧ p = 2^k * q + 1} := by⊢ True ↔ {p | ∃ q k, Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * q + 1}.Infinite
  sorryAll goals completed! 🐙

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

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1065.parts.ii : answer(sorry) ↔
    Set.Infinite {p | ∃ q k l, p.Prime ∧ q.Prime ∧ p = 2^k * 3^l * q + 1} := by⊢ True ↔ {p | ∃ q k l, Nat.Prime p ∧ Nat.Prime q ∧ p = 2 ^ k * 3 ^ l * q + 1}.Infinite
  sorryAll goals completed! 🐙


end Erdos1065