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.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement restricted_prime_number_theorem as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `restricted_prime_number_theorem`	research open	Formal Conjectures
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 2026-11-11). 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. 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 2026-11-12}
}

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Claims

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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

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».

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