Balanced prime conjecture

Let p_k be the k-th prime number. Are there infinitely many n such that (p_n + p_n+2) / 2 is prime?

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-balanced-primes,
  title        = {Balanced prime conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/balanced-primes}},
  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

balanced_primes. Let $p_{k}$ be the $k$ -th prime number. Are there infinitely many $n$ such that $(p_{n} + p_{n + 2}) /2$ is prime?

balanced_primes_order. Let $p_{k}$ be the $k$ -th prime number. Are there infinitely many $n$ such that $p_{n} = \frac{\sum _{i = 1}^{k} p _{n - i} + p _{n + i}}{2 * k}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.BalancedPrimes (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem balanced_primes :
    answer(sorry) ↔ {n : ℕ | Prime ((Nat.nth Prime n + Nat.nth Prime (n + 2)) / 2)}.Infinite

theorem balanced_primes_order :
    answer(sorry) ↔ ∀ k > 0, {n : ℕ | k ≤ n ∧ 2 * k * Nat.nth Prime n = ∑ i ∈ .Ioc 0 k,
      ((n - i).nth Prime + (n + i).nth Prime)}.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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

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