# P2994: Erdős Problem 41 on Distinct Triple Sums

- ID: `P2994`
- Reference: `erdos-problem-41`
- Page: https://theoremdb.org/statements/P2994
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit
\[
\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}}
\]
equal $0$?

### Context

This problem was posed by Paul Erdős and concerns the maximum possible density of an infinite set of natural numbers with all triple sums distinct. The analogous question for pairwise distinct sums, where the exponent $1/3$ is replaced by $1/2$, was proved by Erdős.

### Problem setup

- **Definition (Definition 1).** A set $A \subseteq \mathbb{N}$ satisfies the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ each of cardinality $3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies that $I = J$.
- **Definition (For a set $S$, the notation $|S|$).** For a set $S$, the notation $|S|$ denotes the cardinality of $S$.
- **Definition (For a sequence $(a_N)_{N \geq 1}$ of real numbers, the lower limit $\liminf_{N \to \infty} a_N$).** For a sequence $(a_N)_{N \geq 1}$ of real numbers, the lower limit $\liminf_{N \to \infty} a_N$ is defined as $\lim_{N \to \infty} \inf_{n \geq N} a_n$, which equals the limit of the infimum of the tail of the sequence.
- **Remark.** This problem was posed by Paul Erdős and concerns the maximum possible density of an infinite set of natural numbers with all triple sums distinct. The analogous question for pairwise distinct sums, where the exponent $1/3$ is replaced by $1/2$, was proved by Erdős.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit
\[
\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}}
\]
equal $0$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 41 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit
\[
\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}}
\]
equal $0$? [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 41 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit
\[
\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}}
\]
equal $0$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 41 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 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit
\[
\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}}
\]
equal $0$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 41 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 41 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 41: A set $A \subseteq \mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \subseteq A$ with $|I| = |J| = 3$, the equality $\sum_{i \in I} i = \sum_{j \in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \subseteq \mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \cap \{1, \ldots, N\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit \[ \liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{N^{1/3}} \] equal $0$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-41`, 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 41, maintained status record. Erdős Problems record 41, checked 2026-08-01. Problem 41; status field and linked bibliography https://www.erdosproblems.com/41
   - Also cited at Problem 41; 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 Erdős Problem 41 on Distinct Triple Sums: 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 41 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 Erdős Problem 41 on Distinct Triple Sums: 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 41. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/41.lean:L44; theorem erdos_41; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/41.lean#L44
   - 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 Erdős Problem 41 on Distinct Triple Sums: Supplies the pinned formal declaration whose human-readable editorial statement is published here.