[#P2970] Convergence of an alternating series involving primes

Problem. Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?

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Definitions and notation

1Context

This problem is one of several questions posed by Paul Erdős concerning the behavior of series involving prime numbers. The alternating sign and the ratio of the index to the corresponding prime create delicate convergence properties that are not immediately resolved by standard tests.

2Definitions

Definition 1 (The sequence of prime numbers \(p_n\). The sequence of prime numbers \(p_n\) is the increasing enumeration of all prime natural numbers, beginning with \(p_1 = 2\).

Definition 2 (A series \(\sum_{n=1}^{\infty} a_n\) of real (or rational) numbers). A series \(\sum_{n=1}^{\infty} a_n\) of real (or rational) numbers is convergent if the sequence of partial sums \(S_N = \sum_{n=1}^{N} a_n\) approaches a finite limit as \(N \to \infty\).

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?^[1] Relevance to this problem For erdos problem 15, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent?^[1] Relevance to this problem For Convergence of an alternating series involving primes, record erdos-problem-15-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 15: Let \(p_n\) denote the \(n\)-th prime number, with \(p_1 = 2, p_2 = 3, p_3 = 5,\ldots\). Consider the infinite series \[\sum_{n=1}^{\infty} (-1)^n \frac{n}{p_n}.\] Is this series convergent? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 15 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 15 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemConvergence of an alternating series involving primes

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Convergence of an alternating series involving primes.” TheoremDB. P2970. Problem statement; statement text SHA-256 4bc582c3068aca7b5907333ba2e28724a1f34b92bd392221917448025e0ca2b3. https://theoremdb.org/statement/?ref=P2970

BibTeX

@misc{theoremdb-problem-4bc582c3068aca7b5907333ba2e28724a1f34b92bd392221917448025e0ca2b3,
  title = {{Convergence of an alternating series involving primes}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4bc582c3068aca7b5907333ba2e28724a1f34b92bd392221917448025e0ca2b3},
  url = {https://theoremdb.org/statement/?ref=P2970}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 15, maintained status record. Erdős Problems record 15, checked 2026-08-01. Problem 15; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 15; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Convergence of an alternating series involving primes: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 15 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Convergence of an alternating series involving primes: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 15. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/15.lean:L37; theorem erdos_15; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Convergence of an alternating series involving primes: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P2970] Convergence of an alternating series involving primes

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