arXiv subjects

Basile Beyer de Ryke

Publications and source records attributed to Basile Beyer de Ryke.

2 recordsLinked to original sources

arXiv2026

The solution to Kourovka problem 21.88

We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.

math.GR↗

arXiv2026

A density deficit for sums of three cube-full numbers

Let $F$ be the set of positive cube-full integers. We prove that, for every $\varepsilon>0$, there is a reduced residue class in which $F+F+F$ has upper relative density at most $\varepsilon$. Consequently, the positive integers not representable as sums of at most three cube-full numbers have positive lower natural density. This proves the infinitude assertion in Erdős Problem \#940 for $r=3$. The argument combines a cubic-character estimate with a truncation of the parametrisation by the canonical 4-full part.

math.GM↗