Problem 1102

We say that $A\subseteq \mathbb{N}$ has property $P$ if, for all $n\geq 1$ , there are only finitely many $a\in A$ such that $n+a$ is squarefree.

We say that $A$ has property $Q$ if there are infinitely many $n$ such that $n+a$ is squarefree for all $a<n$ .

How fast must sequences $A=\{a_1<a_2<\cdots\}$ with properties $P$ or $Q$ increase?

Second statement

We say that $A\subseteq \mathbb{N}$ has property $P$ if, for all $n\geq 1$ , there are only finitely many $a\in A$ such that $n+a$ is squarefree.

We say that $A$ has property $Q$ if there are infinitely many $n$ such that $n+a$ is squarefree for all $a\in A$ with $a<n$ .

How fast must sequences $A=\{a_1<a_2<\cdots\}$ with properties $P$ or $Q$ increase?

Source: erdosproblems.com/1102

Number theory

Wiki page

Status

SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Labeled SOLVED (LEAN) on the site. The standing is solved, derived from the accepted full claim of van Doorn and Tao [vDTa25], published in Acta Arithmetica in 2026, on its claim page (Van Doorn and Tao, 2025). The paper answers the corrected Statement: a sequence with property $P$ must have natural density zero, and nothing more, since the density may tend to zero arbitrarily slowly; a sequence with property $Q$ has upper density at most $6/\pi^2$ , and a squarefree sequence with property $Q$ and density exactly $6/\pi^2$ exists; an admissible sequence (one avoiding a residue class modulo $p^2$ for every prime $p$ ) with $a_j\ge\exp(Cj/\log j)$ for infinitely many $j$ has property $Q$ , so $2^n\pm1$ and $n!\pm1$ do, while fast growth alone does not suffice. The site's curator, Thomas Bloom, credits the paper with the result, the reviewed evidence recorded on the claim page. The site's (LEAN) qualifier refers to the first author's Lean files of 23 February 2026, produced by Aristotle, Harmonic's prover, two of which the formal-conjectures catalog registers; this corpus has not built them. Whether $2^n\pm1$ or $n!\pm1$ has property $P$ , a side question of the site's commentary, remains unanswered. The commentary (last edited 2 December 2025) and the thread were accessed 2026-10-07.

Notes

The site's definition of property $Q$ quantifies over every $a<n$ , not over the members of $A$ , and so read it holds for no set $A$ : for $n\ge5$ the numbers $n+1,\ldots,2n-1$ are at least four consecutive integers, one of them is a multiple of $4$ , so only $n\le4$ can qualify, and the question about sequences with property $Q$ concerns an empty class (an elementary check, the corpus's own). The change replaces "for all $a<n$ " by "for all $a\in A$ with $a<n$ "; nothing else changes. The evidence is Erdős's own definition [Er81h, printed p. 179]: " $A$ is said to have property $Q$ if for infinitely many $n$ , $n+a_i$ is squarefree for all $a_i<n$ ", where the $a_i$ are the terms of $A=\{a_1<a_2<\cdots\}$ , followed by "It is easy to see that if $A$ increases sufficiently fast then it has property $Q$ ", which is true only when the condition ranges over the members of $A$ . The inserted words are those of Definition 1 of [vDTa25], which recalls Erdős's definitions and states property $Q$ as " $n+a$ squarefree for all $a\in A$ with $a<n$ ". The defect is the site's: Erdős's text carries the membership in the subscript of $a_i$ . No result about the site's wording is recorded.

Problem 1102

Second statement

Status

Notes

Proof claims

Discussion