Tug of war score between prime gap increases and decreases

Is the score a(n) > 0 for some n > 250000?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-92243,
  title        = {Tug of war score between prime gap increases and decreases},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-92243}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Is the score $a (n) > 0$ for some $n > 250000$ ?

conjecture2. Is the score $a (n)$ bounded from below?

conjecture3. Is the score $a (n)$ bounded from above?

conjecture4. Is the score $a (n) > 0$ for infinitely many values of $n$ ?

conjecture5. Is the score $a (n) < 0$ for infinitely many values of $n$ ?

Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases: start with score $= 0$ at $n = 1$ and at stage $k > 1$ , increase (resp. decrease) the score by $1$ if the $k$ -th prime gap is greater (resp. less) than the previous prime gap.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«92243» (5 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture1 : answer(sorry) ↔ ∃ n > 250000, a n > 0

theorem conjecture2 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, B ≤ a n

theorem conjecture3 : answer(sorry) ↔ ∃ B : ℤ, ∀ n : ℕ, a n ≤ B

theorem conjecture4 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n > 0}

theorem conjecture5 : answer(sorry) ↔ Set.Infinite {n : ℕ | a n < 0}

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A092243

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.