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Claim. Theorem 1.2 of B. S. Ho, A squarefree lower bound for Erdős Problem 675 (5 pp.). For each $0<c<25/72$ there is $N_0(c)$ such that, for $N\ge N_0(c)$, every $t\ge1$ with $1_S(a+t)=1_S(a)$ for $1\le a\le N$, where $S$ is the set of squarefree positive integers, is divisible by $\prod_{p\le N^c}p^2$. Hence $t>\exp(N^c)$, and this holds for the least such $t$, whose existence is the note's Proposition 1.1. When $p^2\nmid t$ the proof finds a squarefree $a\le N$ with $p^2\mid a+t$, using Nunes's bound $L(q,r)\ll_\varepsilon q^{36/25+\varepsilon}$ for the least squarefree integer in a reduced class (Mathematika 63 (2017), Corollary 1.2) at $q=p$ and $q=p^2$; the prime number theorem then gives the size. The note credits GPT-5.4 Pro with finding the proof and the author with checking it.

Covers. The yes-or-no part of the squarefree question of Problem 675 (the page's third question): $t_n>\exp(n^c)$ for some $c>0$, answered yes for all large $n$ and every $c<25/72$. For small $n$ the inequality fails ($t_1=1$), so "all large $n$" is the reading of Erdős's expectation that $t_n>\exp(n^c)$. How fast the least shift grows is addressed only through this lower bound. The first two questions are not addressed.

Depends on. No page of this wiki. The proof rests on Nunes's Corollary 1.2, which has no library page, and on the prime number theorem.

Standing. Claimed. An automated check posted in the thread on 19 April 2026 reported no issue, and Terence Tao replied on 28 April 2026 that it looked good to Tao. Both are commentary on a problem the site labels OPEN, not acceptance. There is no refereed or formalized version.