Are prime numbers among sums of prime numbers distributed as frac n2ln(n)?

The conjecture claims that π_n∼frac n2ln(n). In other words, primes are distributed among the much sparser sequence (S_n)_n with essentially the same density as in the positive integers, up to a factor of 2. MathOverflow 434111.

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-mathoverflow-434111,
  title        = {Are prime numbers among sums of prime numbers distributed as frac n2ln(n)?},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mathoverflow-434111}},
  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

The conjecture claims that $π_{n} \sim \frac{n}{2 l n ( n )}$ .

In other words, primes are distributed among the much sparser sequence $(S_{n})_{n}$ with essentially the same density as in the positive integers, up to a factor of $2$ .

MathOverflow 434111.

[Me18] Meštrović, R., Curious Conjectures on the Distribution of Primes Among the Sums of the First 2n Primes, arXiv:1804.04198 (2018).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Mathoverflow.«434111». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem restricted_prime_number_theorem :
    answer(sorry) ↔ ((fun n : ℕ => (piRestricted n : ℝ)) ~[atTop] (fun n : ℕ => (n : ℝ) / (2 * Real.log n)))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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.

Variants

restricted_prime_number_theorem.variants.even_subsequence — Meštrović's original formulation [Me18, Conjecture 3.3]: the sequence of sums of the first 2m primes satisfies the Restricted Prime Number Theorem, π(m…

References

mathoverflow.net/questions/434111

[Me18] Meštrović, R., Curious Conjectures on the Distribution of Primes Among the Sums of the First 2n Primes, arXiv:1804.04198 (2018), Conjecture 3.3.

Source and licence

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