Erdős Problem #1209

The question

erdos_1209.parts.iii.b. Are there $n$ such that $n+2^{2^k}$ is always squarefree?

erdos_1209.parts.iii.c. Are there $n$ such that $n+2^{2^k}$ is infinitely often a prime?

erdos_1209.parts.iii.d. Are there $n$ such that $n+2^{2^k}$ is infinitely often squarefree?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1209» (3 statements).

theorem erdos_1209.parts.iii.b :
    ∃ n : ℕ, ∀ k : ℕ, Squarefree (n + 2 ^ (2 ^ k))

theorem erdos_1209.parts.iii.c :
    ∃ n : ℕ, {k | (n + 2 ^ (2 ^ k)).Prime}.Infinite

theorem erdos_1209.parts.iii.d :
    ∃ n : ℕ, {k | Squarefree (n + 2 ^ (2 ^ k))}.Infinite

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1209. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/429
• erdosproblems.com/1102
• erdosproblems.com/1209
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.