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?

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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).

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

theorem balanced_primes_order :
    ∀ 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.