Problem 15

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claims/: The 1 claim page of Problem 15, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that

$$\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$$

converges, where $p_n$ is the sequence of primes?

Status. Open; the site's label is OPEN. The best result is conditional: Tao proves that the series converges assuming a quantitative Hardy-Littlewood prime tuples conjecture (Theorem 1.4 of arXiv:2308.07205, published as Comm. Amer. Math. Soc. 4 (2024), 80-96); the result is recorded as the accepted conditional claim page Tao 2023, which derives no standing. A Lean file accepted by the bounty site Conjectures.io (record f8fbf2ed-4ae2-49ab-b0b0-f2f7924af6b4, solution, accessed 2026-09-28; kernel verified; review outcome a formalization-defect award under its policy v1, whose decision file is dated 2026-08-05 and which Conjectures.io displays as decided 25 August 2026; certified 6 August 2026) proves the negation of the formal-conjectures statement as it stood from 2026-04-17 to 2026-09-09, True ↔ Summable (fun k : ℕ => (-1 : ℚ) ^ (k + 1) * (k + 1) / (k.nth Nat.Prime)). That statement is not the site's question: Mathlib's Summable is unconditional summability, over the reals equivalent to absolute convergence, and its rational coefficients demand a rational limit, so the file refutes the absolute-convergence variant ($\sum n/p_n$ diverges since $n/p_n\sim 1/\log n$) and says nothing about the partial sums. Conjectures.io's review records that the result does not settle the informal Erdős problem, and Conjectures.io withdrew the problem from its pool on 5 August 2026 pending a corrected statement; formal-conjectures restated the theorem as convergence of the real partial sums on 9 September 2026 (PR #4990) and keeps it open. No proof claim on erdosproblems.com, no refereed resolution and no other candidate was found. The Conjectures.io submission has no claim page of its own: its submitter is pseudonymous on Conjectures.io, and its own decision classes it as the refutation of a defective formal task that settles nothing about the problem.

Source. erdosproblems.com/15, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #15, https://www.erdosproblems.com/15.

References.

• [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
• [Ta23] Tao, T., The convergence of an alternating series of Erdős, assuming the Hardy-Littlewood prime tuples conjecture. arXiv:2308.07205 (2023); Comm. Amer. Math. Soc. 4 (2024), no. 3, 80-96, DOI 10.1090/cams/29.
• [Zh14] Zhang, Yitang, Bounded gaps between primes. Ann. of Math. (2) (2014), 1121-1174.

Formalization. Statement in formal-conjectures, at the linked commit and stated since 2026-09-09 (PR #4990) as the existence of a real limit of the partial sums; the earlier statement (2026-04-17 to 2026-09-09) used Summable over ℚ, which is absolute convergence with a rational limit, and was refuted on Conjectures.io as a formalization defect (see Status).

Current assessment

The site formulation of 2026-09-27 (history page: one prior revision dated 2025-10-20) asks whether the partial sums of $\sum(-1)^n n/p_n$ converge; the wording is not defective. Status open: convergence is known only under Tao's quantitative Hardy-Littlewood hypothesis (Theorem 1.4, refereed in Comm. Amer. Math. Soc. 4 (2024), 80-96), and the absolute series diverges. The Conjectures.io acceptance of 5-6 August 2026 refutes a misformalization (Summable over ℚ) and Conjectures.io's own review says it settles nothing; formal-conjectures corrected the statement on 2026-09-09.

Dated search scope, 2026-09-27: erdosproblems.com (the problem page, the forum thread /forum/discuss/15 with two comments dated 11 August 2025 and 13 January 2026 and no proof claim, the proof-claims thread, and the history page); the community database (teorth/erdosproblems problems.yaml, entry 15: status open, last update 2025-08-31); conjectures.io (the results list, the record, its problem and solution pages, the report API, the validator's review-decision file, and the task pool listing at conjectures-io/conjectures-tasks pool/tier-1, whose directory listing held no problem-15 task); formal-conjectures (15.lean at main and at a commit carrying the statement quoted in Status, its history, issue #4979, and PR #4990); arXiv (the 2308.07205 version history, v3 of 23 August 2023 being the latest, and export-API searches for the alternating series); and Crossref (10.1090/cams/29). No proof claim, preprint or acceptance of a resolution was found.

Reviewed coverage: the four reconstruction pages of Tao's conditional argument (Theorem 1.4 with Lemmas 3.1 and 3.2 and the Section 2 equivalence), filed with Conjecture 1.3 stated as the imported hypothesis under the Problem 15 research folder, each received a focused independent review on 2026-09-28, graded separately: fidelity to the source faithful on all four; the arguments of Lemma 3.1 and relation (2.1) sound, that of Lemma 3.2 sound after the corrections the review asked for; and the Theorem 1.4 argument sound conditional on Conjecture 1.3. The corrections were applied. No tier is assigned, the reviews are not acceptance evidence, and they cover the reconstruction, not Tao's published text, which the library records as a digest; the accepted Lean file was not built here. The problem's one claim page, Tao 2023, records the conditional theorem as accepted on its refereed publication; being conditional, it derives no standing, and the problem stays open.

Progress

Convergence is known only conditionally; see the Current assessment above and the Known Results below.

Known Results

• Conditional convergence: Theorem 1.4 of Tao 2023 (arXiv:2308.07205v3, published as Comm. Amer. Math. Soc. 4 (2024), 80-96, DOI 10.1090/cams/29) proves that, assuming Conjecture 1.3, a quantitative Hardy-Littlewood prime tuples conjecture with power-saving error uniform for $k\le(\log\log x)^5$ and shifts in $[0,\log^2 x]$, the series $\sum(-1)^n n/p_n$ converges. The proof goes through Said's equivalence (Section 2) with convergence of $\sum_{n\ge2}(-1)^{\pi(n)}/(n\log n)$, the van der Corput A-process, and the Banks-Ford-Tao random sifted model; numerical computation (Tao, p. 1) suggests slow convergence to roughly $-0.052161$. Unconditionally the question is open.
• The absolute series $\sum n/p_n$ diverges ($n/p_n\sim 1/\log n$ and $\sum 1/p_n$ diverges), so the question is one of conditional convergence only. This is the content of the Lean file accepted by Conjectures.io on 5-6 August 2026 (record f8fbf2ed-4ae2-49ab-b0b0-f2f7924af6b4), which proves the negation of the formal-conjectures statement quoted in the Status field: Mathlib Summable is unconditional summability (absolute convergence over the reals) and the rational coefficients demand a rational limit, so the refuted statement is a variant strictly stronger than the catalog question. Conjectures.io's review classed the acceptance as a formalization-defect award and states that the result must not be described as a solution or counterexample to the informal problem; formal-conjectures corrected the statement to convergence of the real partial sums on 2026-09-09 (PR #4990). Settles no part of the catalog question.
• Companion series from [Er98] (site remarks, not the catalog question): $\sum(-1)^n/(p_{n+1}-p_n)$ diverges, because the bounded gaps of Zhang 2014 give infinitely many terms of absolute value at least a fixed constant (an observation the site credits to Weisenberg); a forum comment of 11 August 2025 sketches, assuming the prime $k$-tuples conjecture, that this series is unbounded in at least one direction (an unrefereed sketch). Erdős and Nathanson reported, and the site records a Selberg-sieve argument it credits to Sawhney, that $\sum(-1)^n/(n(p_{n+1}-p_n)(\log\log n)^c)$ converges absolutely for $c>2$, while Erdős conjectured convergence for every $c>0$.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• tao_2023_convergence_alternating_series_erdos_assuming_hardy
• zhang_2014_bounded_gaps_between_primes
• zhang_2014_bounded_gaps_between_primes / theorem_1
• zhang_2014_bounded_gaps_between_primes / theorem_2