1. FormalConjectures
  2. ErdosProblems
  3. «244»
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Erdős Problem 244

Reference: erdosproblems.com/244

namespace Erdos244

Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_244 : answer(sorry) ↔ ∀ C > (1 : ℝ), 0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity := ⊢ True ↔ ∀ C > 1, 0 < {x | ∃ p k, ∃ (_ : Nat.Prime p), p + ⌊C ^ k⌋₊ = x}.lowerDensity All goals completed! 🐙

Romanoff [Ro34] proved that the answer is yes if $C$ is an integer.

[Ro34] Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.

@[category research solved, AMS 11] theorem declaration uses 'sorry'erdos_244.variants.Romanoff {C : ℕ} (hC : 1 < C) : 0 < { p + ⌊C ^ k⌋₊ | (p) (k) (_ : p.Prime) }.lowerDensity := C:ℕhC:1 < C⊢ 0 < {x | ∃ p k, ∃ (_ : Nat.Prime p), p + ⌊C ^ k⌋₊ = x}.lowerDensity All goals completed! 🐙 end Erdos244