# P3052: Existence of the Mean Power Limit for Squarefree Gaps

- ID: `P3052`
- Reference: `erdos-problem-145`
- Page: https://theoremdb.org/statements/P3052
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
\[
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}
\]
as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit
\[
\lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha}
\]
exist for all $\alpha \geq 0$?

### Context

This problem concerns the distribution of gaps between consecutive squarefree numbers. A squarefree number is a positive integer that is not divisible by any perfect square other than $1$. Erdős originally posed this question about whether the mean value of powers of gaps between consecutive squarefree numbers has a limiting value. Partial results are known: the limit exists for $0 \leq \alpha \leq 2$ (Erdős, 1951), for $0 \leq \alpha \leq 3$ (Hooley, 1973), and for $0 \leq \alpha \leq 11/3$ (Greaves, Harman, and Huxley, 1997).

### Problem setup

- **Definition (A positive integer $n$).** A positive integer $n$ is called squarefree if no prime squared divides $n$, or equivalently, if in the prime factorization of $n$, every prime appears with exponent at most $1$.
- **Definition (The sequence $(s_n)_{n \geq 1}$ of squarefree numbers).** The sequence $(s_n)_{n \geq 1}$ of squarefree numbers is the strictly increasing enumeration of all squarefree positive integers.
- **Definition (For a real number $x \geq 0$, the set $A(x)$ consists of all indices $n$ such that $s_n \leq x$).** For a real number $x \geq 0$, the set $A(x)$ consists of all indices $n$ such that $s_n \leq x$.
- **Definition (For a real number $\alpha \geq 0$ and a real number $x > 0$, the expression $\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}$).** For a real number $\alpha \geq 0$ and a real number $x > 0$, the expression $\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}$ is the average of the $\alpha$th powers of gaps between consecutive squarefree numbers, where the average is taken over all gaps starting at squarefree numbers up to $x$.
- **Remark.** This problem concerns the distribution of gaps between consecutive squarefree numbers. A squarefree number is a positive integer that is not divisible by any perfect square other than $1$. Erdős originally posed this question about whether the mean value of powers of gaps between consecutive squarefree numbers has a limiting value. Partial results are known: the limit exists for $0 \leq \alpha \leq 2$ (Erdős, 1951), for $0 \leq \alpha \leq 3$ (Hooley, 1973), and for $0 \leq \alpha \leq 11/3$ (Greaves, Harman, and Huxley, 1997).

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
\[
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}
\]
as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit
\[
\lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha}
\]
exist for all $\alpha \geq 0$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 145 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
\[
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}
\]
as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit
\[
\lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha}
\]
exist for all $\alpha \geq 0$? [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 145 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
\[
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}
\]
as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit
\[
\lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha}
\]
exist for all $\alpha \geq 0$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 145 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 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity
\[
\frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha}
\]
as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit
\[
\lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha}
\]
exist for all $\alpha \geq 0$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 145 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 145 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 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \leq x$. For each real number $\alpha \geq 0$, consider the quantity \[ \frac{1}{x} \sum_{n \in A(x)} (s_{n+1} - s_n)^{\alpha} \] as $x \to \infty$. Does there exist, for every $\alpha \geq 0$, a real number $\beta$ such that this quantity converges to $\beta$ as $x \to \infty$? Equivalently, does the limit \[ \lim_{x \to \infty} \frac{1}{x} \sum_{s_n \leq x} (s_{n+1} - s_n)^{\alpha} \] exist for all $\alpha \geq 0$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-145`, 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 145, maintained status record. Erdős Problems record 145, checked 2026-08-01. Problem 145; status field and linked bibliography https://www.erdosproblems.com/145
   - Also cited at Problem 145; 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 Existence of the Mean Power Limit for Squarefree Gaps: 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 145 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 Existence of the Mean Power Limit for Squarefree Gaps: 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 145. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/145.lean:L46; theorem erdos_145; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/145.lean#L46
   - 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 Existence of the Mean Power Limit for Squarefree Gaps: Supplies the pinned formal declaration whose human-readable editorial statement is published here.