[#P2994] Erdős Problem 41 on Distinct Triple Sums

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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 $0$?

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Definitions and notation

1Context

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.

2Definitions

Definition 1 (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 2 (For a set $S$, the notation $|S|$). For a set $S$, the notation $|S|$ denotes the cardinality of $S$.

Definition 3 (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.

3What 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 $0$?

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 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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 $0$?	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 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] Relevance to this problem For erdos problem 41, pins the dated research frontier: 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. 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 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 $0$? Open full record →
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 $0$?	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 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] Relevance to this problem For Erdős Problem 41 on Distinct Triple Sums, record erdos-problem-41-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 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 $I = J$. 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 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 $0$? 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 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.

How the 2 records connect

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

ProblemErdős Problem 41 on Distinct Triple Sums

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

“Erdős Problem 41 on Distinct Triple Sums.” TheoremDB. P2994. Problem statement; statement text SHA-256 94a77ee929700743df789ee106598d3a09acb79456ee6c49405be69bb8a8e709. https://theoremdb.org/statement/?ref=P2994

BibTeX

@misc{theoremdb-problem-94a77ee929700743df789ee106598d3a09acb79456ee6c49405be69bb8a8e709,
  title = {{Erdős Problem 41 on Distinct Triple Sums}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 94a77ee929700743df789ee106598d3a09acb79456ee6c49405be69bb8a8e709},
  url = {https://theoremdb.org/statement/?ref=P2994}
}

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.

1References

Packet source. Erdős Problems database, Problem 41, maintained status record. Erdős Problems record 41, checked 2026-08-01. Problem 41; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 41 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 Erdős Problem 41 on Distinct Triple Sums: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 41. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/41.lean:L44; theorem erdos_41; 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 Erdős Problem 41 on Distinct Triple Sums: 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.

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[#P2994] Erdős Problem 41 on Distinct Triple Sums

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