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Combinatorics

Erdős 535 - first open case

The first open case of Erdős Problem 535 is r=3r = 3: there should exist c>0c > 0 such that f3(N)Nc/loglogNf_3(N) \leq N^{c/\log\log N} for all sufficiently large NN.

References

  • Er64 P. Erdős, On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644–646.
  • AbHa70 H. L. Abbott and D. Hanson, An extremal problem in number theory. Bull. London Math. Soc. (1970), 324–326.
  • Er73 P. Erdős, Problems and results on combinatorial number theory, in *A Survey of Combinatorial Theory*, North-Holland, 1973.

No one has attempted this yet.

Formal statement

Lean type

∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f 3 N) ≤ ↑N ^ (c / Real.log (Real.log ↑N))

What you must prove

import FormalConjectures.ErdosProblems.«535»
import TaskSupport

namespace Bounty

theorem target : fcTypeOfName% "Erdos535.erdos_535.variants.first_open_case" := by
  sorry

end Bounty

Pinned source: FormalConjectures/ErdosProblems/535.lean

Source type SHA-256
sha256:4940f37b482af2f8dde043c96224ee6c7df8b9909463615304905090c881a800
Task id
fc-379fc029-variants-first-open-case-09c89f4ec6-formalized-v1
Task commitment
sha256:22cf446ce24c40fb947c22ab29e662d3b26f7002ad6573aed8df204733b81c00

Something wrong with this formalization?

A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.