# P2966: Convergence of the Reciprocal Sum for Good Sets

- ID: `P2966`
- Reference: `erdos-problem-12`
- Page: https://theoremdb.org/statements/P2966
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges.

### Context

This problem is part of a family of questions posed by Erdős concerning the density and structure of sets of natural numbers with restricted divisibility properties. Erdős and Sárközy proved that any good set must have asymptotic density zero. It is known that there exist good sets $A$ for which $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/2}} > 0$, and that the set of squares of primes $p \equiv 3 \pmod{4}$ provides an example with this property. The question of whether the reciprocal sum must always converge remains open.

### Problem setup

- **Definition (A set $A \subseteq \mathbb{N}$).** A set $A \subseteq \mathbb{N}$ is good if $A$ is infinite and there do not exist three distinct elements $a, b, c \in A$ such that $a \mid (b+c)$, $b > a$, and $c > a$.
- **Definition (For a set $A \subseteq \mathbb{N}$, the sum $\sum_{n \in A} \frac{1}{n}$).** For a set $A \subseteq \mathbb{N}$, the sum $\sum_{n \in A} \frac{1}{n}$ is said to converge (or be summable) if the series of reciprocals of elements of $A$, taken in increasing order, has a finite limit.
- **Remark.** This problem is part of a family of questions posed by Erdős concerning the density and structure of sets of natural numbers with restricted divisibility properties. Erdős and Sárközy proved that any good set must have asymptotic density zero. It is known that there exist good sets $A$ for which $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/2}} > 0$, and that the set of squares of primes $p \equiv 3 \pmod{4}$ provides an example with this property. The question of whether the reciprocal sum must always converge remains open.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 12 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges. [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 12 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 12 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 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 12 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 12 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 12: A set $A$ of natural numbers is called good if it is infinite and there do not exist distinct elements $a, b, c \in A$ such that $a$ divides $b + c$ with $b > a$ and $c > a$. Determine whether the following statement is true: For every good set $A \subseteq \mathbb{N}$, the series $\sum_{n \in A} \frac{1}{n}$ converges. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-12`, 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 12, maintained status record. Erdős Problems record 12, checked 2026-08-01. Problem 12; status field and linked bibliography https://www.erdosproblems.com/12
   - Also cited at Problem 12; 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 the Reciprocal Sum for Good Sets: 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 12 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 the Reciprocal Sum for Good Sets: 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 12. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/12.lean:L82; theorem erdos_12.parts.iii; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/12.lean#L82
   - 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 the Reciprocal Sum for Good Sets: Supplies the pinned formal declaration whose human-readable editorial statement is published here.