Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1149

claims/: The 3 claim pages of Problem 1149, one per claimant's result; the problem's standing derives from them.

Statement. Let $\alpha>0$ be a real number, not an integer. The density of integers $n\geq 1$ for which $(n,\lfloor n^\alpha\rfloor)=1$ is $6/\pi^2$ .

Status. Proved. Theorem 1 of Bergelson and Richter ([BeRi17], a chapter of a Springer Festschrift volume, arXiv:1611.08044) gives density $6/\pi^2$ for the integers $n$ coprime to $\lfloor f(n)\rfloor$ for every Hardy-field $f$ meeting its growth conditions, $f(t)=t^\alpha$ with $\alpha>0$ a non-integer among them; the site's curator, Thomas Bloom, records this as the proof. The claim page Bergelson and Richter 2017 records the acceptance. Earlier, Delmer and Deshouillers (2002) proved the statement for every non-integer $\alpha>0$ , Delmer and Deshouillers 2002, and Lambek and Moser (1955) proved the exponents $\alpha=1/k$ for integers $k\ge2$ , Lambek and Moser 1955. Bergelson and Richter's introduction credits Lambek and Moser with all $0<\alpha<1$ , which is more than their paper states.

Source. erdosproblems.com/1149, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1149, https://www.erdosproblems.com/1149.

References.

[BeRi17] Bergelson, Vitaly and Richter, Florian Karl, On the density of coprime tuples of the form $(n,\lfloor f_1(n)\rfloor,\dots,\lfloor f_k(n)\rfloor )$, where $f_1,\dots,f_k$ are functions from a Hardy field. (2017), 109-135; in Number Theory -- Diophantine Problems, Uniform Distribution and Applications (Festschrift for Robert F. Tichy), Springer, DOI 10.1007/978-3-319-55357-3_5; arXiv:1611.08044. Library home: bergelson_2017_density_coprime_tuples_form_where_are.
[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

Linked from (8)

Lambek and Moser's coprimality density for n and [n^{1/k}]Delmer and Deshouillers's coprimality density for n and [n^c]Bergelson and Richter's coprimality density problems/integer_sequences/E1149/claims Sequences and Densities of Integers integer_sequences/bergelson_2017_density_coprime_tuples_form_where_are Theorem 1 (p. 3): gcd(n, ⌊f(n)⌋) = 1 with density 6/π² for Hardy-field f Theorem 2 (p. 4): gcd(n, ⌊f_1(n)⌋, …, ⌊f_k(n)⌋) = 1 with density 1/ζ(k+1)

Graph