[#P3052] Existence of the Mean Power Limit for Squarefree Gaps

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 $\alpha \geq 0$?

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Definitions and notation

1Context

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).

2Problem setup

Definition 1 (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 2 (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.

Remark 1. For a real number $x \geq 0$, the set $A(x)$ consists of all indices $n$ such that $s_n \leq x$.

Definition 3 (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 $x$.

3What 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 $\alpha \geq 0$?

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 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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 $\alpha \geq 0$?	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 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] Relevance to this problem For erdos problem 145, pins the dated research frontier: 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. 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 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 $\alpha \geq 0$? Open full record →
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 $\alpha \geq 0$?	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 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] Relevance to this problem For Existence of the Mean Power Limit for Squarefree Gaps, record erdos-problem-145-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 145: Let $s_1 < s_2 < \cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. 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 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 $\alpha \geq 0$? 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 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.

How the 2 records connect

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

ProblemExistence of the Mean Power Limit for Squarefree Gaps

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

“Existence of the Mean Power Limit for Squarefree Gaps.” TheoremDB. P3052. Problem statement; statement text SHA-256 c88a484ab4317cad8ffbc7cdd5ff64acd81b0060cf8b44a405060c1e6a5b217a. https://theoremdb.org/statement/?ref=P3052

BibTeX

@misc{theoremdb-problem-c88a484ab4317cad8ffbc7cdd5ff64acd81b0060cf8b44a405060c1e6a5b217a,
  title = {{Existence of the Mean Power Limit for Squarefree Gaps}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c88a484ab4317cad8ffbc7cdd5ff64acd81b0060cf8b44a405060c1e6a5b217a},
  url = {https://theoremdb.org/statement/?ref=P3052}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 145, maintained status record. Erdős Problems record 145, checked 2026-08-01. Problem 145; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 145 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 Existence of the Mean Power Limit for Squarefree Gaps: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 145. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/145.lean:L46; theorem erdos_145; 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 Existence of the Mean Power Limit for Squarefree Gaps: 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.

[#P3052] Existence of the Mean Power Limit for Squarefree Gaps

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