# P2970: Convergence of an alternating series involving primes

- ID: `P2970`
- Reference: `erdos-problem-15`
- Page: https://theoremdb.org/statements/P2970
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (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 (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 \(N \to \infty\).
- **Remark.** 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.

### What 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?

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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?

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?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-15`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 15, maintained status record. Erdős Problems record 15, checked 2026-08-01. Problem 15; status field and linked bibliography https://www.erdosproblems.com/15
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 15 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 15. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/15.lean:L37; theorem erdos_15; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/15.lean#L37
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.