Hua Loo Keng's Problem Involving Primes of a Special Type

TL;DR

This paper addresses a variant of Hua Loo Keng's problem focusing on primes related to quadratic irrationalities and fractional parts of their squares, providing a solution for a specific problem involving these primes.

Contribution

It introduces a new approach to Hua Loo Keng's problem involving primes with fractional parts of quadratic irrational multiples, solving a previously open variant.

Findings

Solved the problem for primes with fractional parts of quadratic irrational multiples

Established conditions for the distribution of such primes

Extended understanding of prime distribution related to quadratic irrationals

Abstract

Let $\eta$ be a quadratic irrationality. The variant of Hua Loo Keng's problem involving primes such that $a<\{\eta p^2\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $(0,1)$, solved in this paper.