Year: 2026
Type: preprint
Abstract: Let $\{a_k\}_{k\ge1}$ be an infinite strictly increasing sequence of positive integers with $a_k\mid a_{k+1}$ for every integer $k\ge1$. Dai and Pan proved that the set of integers of the form $p^2+q^2+a_k+a_l$, where $p,q$ are primes and $k,l\ge1$, has positive lower asymptotic density if $\limsup_{k\to\infty}\frac{\log a_k}{k}<\infty$ and $\sup_{k\ge1}\sum_{d\mid a_k}\frac1d<\infty$. They also asked whether the latter condition is necessary. In this paper, we show that this condition is necessary by proving that if $\sup_{k\ge1}\sum_{d\mid a_k}\frac1d=\infty$, then this set has asymptotic density zero.
Source: arXiv (Cornell University)
Authors: Yan, Xiao-Hui, Xu, Ya-Ting
Institution: Anhui Normal University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G1405383360
OpenAlex ID: W7223125013