Problem 378

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claims/: The 1 claim page of Problem 378, one per claimant's result; the problem's standing derives from them.

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Statement. Let $r\geq 0$. Does the density of integers $n$ for which $\binom{n}{k}$ is squarefree for at least $r$ values of $1\leq k<n$ exist? Is this density $>0$?

Status. Proved. The site labels the problem PROVED (page last edited 28 October 2025) and credits Theorem 5 of Granville and Ramaré (Mathematika, 1996), whose bearing on the problem Anay Aggarwal and Stijn Cambie pointed out in the discussion thread in August 2025; the result is recorded on its claim page. Both questions are answered yes. The standing in the frontmatter derives from the claim page.

Source. erdosproblems.com/378, accessed 2026-09-04. The site cites the problem from p. 72 of Erdős and Graham's 1980 problem book. Cite as: T. F. Bloom, Erdős Problem #378, https://www.erdosproblems.com/378.

References.

• [GrRa96] Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43 (1996), no. 1, 73-107. Library home: granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree.

Formalization. None recorded; the statement is not in formal-conjectures and the community database marks the problem unformalized.

Current assessment

The question, as the site states it (page last edited 28 October 2025): for fixed $r\ge0$, do the integers $n$ with at least $r$ squarefree entries $\binom{n}{k}$, $1\le k<n$, have an asymptotic density, and is it positive? Both answers are yes.

What Erdős and Graham knew. The site reports from their 1980 book that, for fixed large $k$, the density of $n$ with $\binom{n}{k}$ squarefree tends to zero as $k$ grows, that infinitely many rows have no squarefree entry between the end ones, and that they expected those rows to have positive density.

The resolution. Granville and Ramaré [GrRa96], Theorem 5, prove that for every $m\ge0$ the rows with exactly $2m+2$ squarefree entries (counting the two ones at the ends) have an asymptotic density $\eta_m$, positive for $m\ge1$ and at most a constant times $\exp(-\tau\sqrt m/\log(2m))$; their Theorem 6 gives, for each fixed $k$, a positive density $c_k$ of $n$ with $\binom{n}{k}$ squarefree, and Theorem 2 shows that squarefree entries sit within $\exp(\tau_1(\log n)^{2/3}(\log\log n)^{1/3})$ of the row's ends. Since the number of squarefree entries in $1\le k<n$ is even once $n>8$, the density the problem asks for is $1-\sum_{2m<r}\eta_m$, and it is at least any $\eta_m$ with $m\ge1$ and $2m\ge r$, hence positive. The paper presents Theorem 5 as the answer to Erdős and Graham's question; Aggarwal (22 August 2025) and Cambie (30 August 2025) brought this to the site, and the curator marked the problem proved. The claim page records the derivation and the acceptance: a refereed paper, credited by the site's curator; no Lean proof is recorded. The same paper's Theorem 1 settles Problem 175.

Search scope, 2026-10-07: the site's problem page, its discussion thread and the community database; the site lists no proof claim for the problem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree
• granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree / theorem_2
• granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree / theorem_5
• granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree / theorem_6
• granville_1996_explicit_bounds_exponential_sums_scarcity_squarefree / theorem_7