Binary Additive Problems Involving Naturals with Binary Decompositions of a Special Kind

TL;DR

This paper derives asymptotic formulas for the number of solutions to specific linear equations involving positive integers with certain binary decomposition constraints, focusing on cases where the variables are bounded by a parameter X.

Contribution

It introduces new asymptotic formulas for counting solutions to binary additive problems with special binary decomposition conditions.

Findings

Asymptotic formulas for solutions to n-3m=h and n-5m=l

Solutions counted for positive integers with m ≤ X

Results applicable to integers with special binary decompositions

Abstract

Let $h$ and $l$ be integers such that $0\le h\le 2$, $0\le l\le 4$. We obtain asymptotic formulas for the numbers of solutions of the equations $n-3m=h$, $n-5m=l$ in positive integers $m$ and $n$ of a special kind, $m\le X$.