[#P2974] Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity

A finite mathematical diagram showing divisors of an integer feeding distinct subset sums. — Divisors of an integer feeding distinct subset sums.

Problem. Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$.

1Context

This is one of three related conjectures posed by Paul Erdős concerning the growth rate of $h(n)$, a measure of how efficiently integers up to $n$ can be represented as sums of divisors of a practical number. A weaker result by Vose (1985) established that $h(m) \ll (\log m)^{1/2}$ for infinitely many practical numbers $m$. The conjecture asks whether the bound can be improved to a power of $\log \log m$.

2Problem setup

Definition 1 (A positive integer $n$). A positive integer $n$ is practical if every integer $m$ with $1 \leq m \leq n$ can be expressed as a sum of distinct positive divisors of $n$.

Definition 2 (For a practical number $n$, the function $h(n)$). For a practical number $n$, the function $h(n)$ is defined as $\max_{1 \leq m \leq n} \min\{|D| : D \subseteq \{\text{positive divisors of } n\}, \, m = \sum_{d \in D} d\}$, i.e., the worst-case minimum number of distinct divisors needed to represent any integer from $1$ to $n$ as a sum of distinct divisors of $n$.

Remark 1. This is one of three related conjectures posed by Paul Erdős concerning the growth rate of $h(n)$, a measure of how efficiently integers up to $n$ can be represented as sums of divisors of a practical number. A weaker result by Vose (1985) established that $h(m) \ll (\log m)^{1/2}$ for infinitely many practical numbers $m$. The conjecture asks whether the bound can be improved to a power of $\log \log m$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$.

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$.^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$.^[1] Relevance to this problem For erdos problem 18, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: 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 18 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 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$. Open full record →
	Trace	Reported
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$.^[1] Relevance to this problem For Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity, record erdos-problem-18-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 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. 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 18: A positive integer $n$ is called practical if every integer $m$ with $1 \leq m \leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \leq m \leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\log \log m)^C$. 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 materialContext, examples, and computations

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 18 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 connectTyped relations and evidence flow

ProblemErdős's Conjecture on Practical Numbers with Bounded Representation Complexity

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-18

This page as plain text: erdos-problem-18.md

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 18, maintained status record. Erdős Problems record 18, checked 2026-08-01. Problem 18; 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 18; 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 Conjecture on Practical Numbers with Bounded Representation Complexity: 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 18 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 Conjecture on Practical Numbers with Bounded Representation Complexity: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 18. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/18.lean:L220; theorem erdos_18a; 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 Conjecture on Practical Numbers with Bounded Representation Complexity: 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.

[#P2974] Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance