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A binary additive equation with prime and square-free number

S. I. Dimitrov

TL;DR

This work addresses the binary additive problem $N=[p^c]+[m^c]$ with $p$ prime and $m$ square-free in the range $1< c<\frac{82}{79}$. The author combines Vaughan's identity, exponent-pair exponential-sum estimates, and a careful decomposition of the generating sum $\Gamma$ into a main positive term and controllable oscillatory terms, establishing a positive lower bound $\Gamma\gg N^{2\gamma-1}$ where $\gamma=1/c$. The secondary contributions are shown to be $O(N^{2\gamma-1}/\log N)$, which suffices to guarantee that representations exist for all sufficiently large $N$. This result extends prior work on primes and square-free numbers in binary additive problems and advances understanding of floor-powers representations in additive number theory.

Abstract

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*} where $p$ is a prime and $m$ is a square-free.

A binary additive equation with prime and square-free number

TL;DR

This work addresses the binary additive problem with prime and square-free in the range . The author combines Vaughan's identity, exponent-pair exponential-sum estimates, and a careful decomposition of the generating sum into a main positive term and controllable oscillatory terms, establishing a positive lower bound where . The secondary contributions are shown to be , which suffices to guarantee that representations exist for all sufficiently large . This result extends prior work on primes and square-free numbers in binary additive problems and advances understanding of floor-powers representations in additive number theory.

Abstract

Let be the floor function. In this paper, we show that when , then every sufficiently large positive integer can be represented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*} where is a prime and is a square-free.
Paper Structure (14 sections, 8 theorems, 107 equations)

This paper contains 14 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notations
  3. Preliminary lemmas
  4. Beginning of the proof
  5. Estimation of $\mathbf{\Gamma_1}$
  6. Lower bound for $\mathbf{\Gamma_3}$
  7. Estimation of $\mathbf{\Sigma_0}$ and $\mathbf{\Sigma_1}$
  8. Consideration of the sum $\mathbf{\textit{W(v)}}$
  9. Application of Vaughan's identity
  10. Estimation of $\mathbf{\Omega_1}$, $\mathbf{\Omega_2}$, $\mathbf{\Omega_3}$ and $\mathbf{\Omega_4}$
  11. Upper bound for $\mathbf{\Sigma_0}$ and $\mathbf{\Sigma_1}$
  12. Lower bound for $\mathbf{\Gamma_1}$
  13. Estimation of $\mathbf{\Gamma_2}$
  14. The end of the proof

Key Result

Theorem 1

Suppose that $1<c<\frac{82}{79}$. Then for every sufficiently large positive integer $N$ the binary equation has a solution in prime number $p$ and square-free number $m$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 5 more