[#P2968] Erdős's Problem on Non-Unique Representability as Sums

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Problem. Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?

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Definitions and notation

1Context

This is part (ii) of Erdős's Problem 14, which investigates how sparse the set of non-uniquely representable integers can be. Part (i) asks whether the complement of $B$ must always be at least of order $N^{1/2-\epsilon}$ for any $\epsilon > 0$. Part (ii) asks whether the stronger bound $o(\sqrt{N})$ is ever achievable.

2Problem setup

Definition 1 (For a set $A \subseteq \mathbb{N}$, the set of all unique sums from $A$). For a set $A \subseteq \mathbb{N}$, the set of all unique sums from $A$ is the set of positive integers that can be written in exactly one way as $a + a'$ with $a, a' \in A$.

Remark 1. For a set $A \subseteq \mathbb{N}$ and a positive integer $N$, the function $f_A(N)$ counts the number of integers in $\{1, 2, \ldots, N\}$ that are not unique sums from $A$.

Remark 2. For functions $f, g: \mathbb{N} \to \mathbb{R}$ with $g$ positive, we write $f = o(g)$ to mean that $f(N)/g(N) \to 0$ as $N \to \infty$; this is the little-o notation.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?

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 14 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?^[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 14 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?	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 14 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?^[1] Relevance to this problem For erdos problem 14, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 14 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For. 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 14 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 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?	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 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$?^[1] Relevance to this problem For Erdős's Problem on Non-Unique Representability as Sums, record erdos-problem-14-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 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. 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 14: For a set $A \subseteq \mathbb{N}$, let $B$ denote the set of positive integers that can be represented in exactly one way as the sum of two (not necessarily distinct) elements of $A$. For each positive integer $N$, define $f_A(N)$ to be the number of integers in $\{1, 2, \ldots, N\}$ that do not belong to $B$; that is, integers that are either not representable at all as a sum of two elements of $A$, or are representable in more than one way. Does there exist a set $A \subseteq \mathbb{N}$ such that $f_A(N) = o(\sqrt{N})$ as $N \to \infty$? Equivalently, is there a set $A \subseteq \mathbb{N}$ for which $f_A(N)/\sqrt{N} \to 0$ as $N \to \infty$? 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 14 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 14 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's Problem on Non-Unique Representability as 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's Problem on Non-Unique Representability as Sums.” TheoremDB. P2968. Problem statement; statement text SHA-256 9408877d116e7299359f004da2b9f200d5c41abcd3e1b8e381a66599f761db69. https://theoremdb.org/statement/?ref=P2968

BibTeX

@misc{theoremdb-problem-9408877d116e7299359f004da2b9f200d5c41abcd3e1b8e381a66599f761db69,
  title = {{Erdős's Problem on Non-Unique Representability as Sums}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 9408877d116e7299359f004da2b9f200d5c41abcd3e1b8e381a66599f761db69},
  url = {https://theoremdb.org/statement/?ref=P2968}
}

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 14, maintained status record. Erdős Problems record 14, checked 2026-08-01. Problem 14; 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 14; 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's Problem on Non-Unique Representability as 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 14 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's Problem on Non-Unique Representability as Sums: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 14. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/14.lean:L58; theorem erdos_14.parts.ii; 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's Problem on Non-Unique Representability as 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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[#P2968] Erdős's Problem on Non-Unique Representability as Sums

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Background

Definitions

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