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

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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 $h(m) < (\log \log m)^C$.

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Definitions and notation

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 $\log \log m$.

2Definitions

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 $n$.

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

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 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

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Record	Kind	Assessment
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 $h(m) < (\log \log m)^C$.	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 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 →
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 $h(m) < (\log \log m)^C$.	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 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 material

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 connect

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

ProblemErdős's Conjecture on Practical Numbers with Bounded Representation Complexity

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 Conjecture on Practical Numbers with Bounded Representation Complexity.” TheoremDB. P2974. Problem statement; statement text SHA-256 c3420d15f61c1e819e9cf1c1e0b1bb2b30ea0c395e1beb4c28b6fb0d1eede6d3. https://theoremdb.org/statement/?ref=P2974

BibTeX

@misc{theoremdb-problem-c3420d15f61c1e819e9cf1c1e0b1bb2b30ea0c395e1beb4c28b6fb0d1eede6d3,
  title = {{Erdős's Conjecture on Practical Numbers with Bounded Representation Complexity}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c3420d15f61c1e819e9cf1c1e0b1bb2b30ea0c395e1beb4c28b6fb0d1eede6d3},
  url = {https://theoremdb.org/statement/?ref=P2974}
}

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 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.

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

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