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

- ID: `P2974`
- Reference: `erdos-problem-18`
- Page: https://theoremdb.org/statements/P2974
- Record maturity: Reviewed problem with recorded work

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

### Context

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 $\log \log m$.

### Problem setup

- **Definition (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 (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.** 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 $\log \log m$.

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

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

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

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-18`, 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 18, maintained status record. Erdős Problems record 18, checked 2026-08-01. Problem 18; status field and linked bibliography https://www.erdosproblems.com/18
   - 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
   - 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'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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 18 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's Conjecture on Practical Numbers with Bounded Representation Complexity: 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 18. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/18.lean:L220; theorem erdos_18a; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/18.lean#L220
   - 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's Conjecture on Practical Numbers with Bounded Representation Complexity: Supplies the pinned formal declaration whose human-readable editorial statement is published here.