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Some results on a conjecture of de Polignac about numbers of the form $p + 2^k$

Yuda Chen, Xiangjun Dai, Huixi Li

Abstract

We have primarily obtained three results on numbers of the form $p + 2^k$. Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form $p + 2^k$, disproving a conjecture by Erdős as Chen did recently. Secondly, we have verified a conjecture by Chen that any arithmetic progression that do not contain numbers of the from $p + 2^k$ must have a common difference which is at least 11184810. Thirdly, we have improved the existing upper bound estimate for the density of numbers that can be expressed in the form $p + 2^k$ to $0.490341088858244$.

Some results on a conjecture of de Polignac about numbers of the form $p + 2^k$

Abstract

We have primarily obtained three results on numbers of the form . Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form , disproving a conjecture by Erdős as Chen did recently. Secondly, we have verified a conjecture by Chen that any arithmetic progression that do not contain numbers of the from must have a common difference which is at least 11184810. Thirdly, we have improved the existing upper bound estimate for the density of numbers that can be expressed in the form to .
Paper Structure (13 sections, 9 theorems, 30 equations, 3 tables, 3 algorithms)

This paper contains 13 sections, 9 theorems, 30 equations, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Constructions of arithmetic progressions not containing numbers of the form $p + 2^k$
  3. Erdős and Chen's constructions
  4. Theoretical framework and algorithm design
  5. Our results and some discussion on Theorem \ref{['apthm']}
  6. Verification of a conjecture by Chen
  7. Improvement on $\overline{d}$
  8. The algorithms by Habsieger and Roblot
  9. Our enhanced algorithm
  10. Enhancements to improve running speed
  11. Enhancements to improve memory efficiency
  12. Utilizing GPU computing to further improve efficiency
  13. Our results and some discussion on Theorem \ref{['upperboundthm']}

Key Result

Theorem 1

The 48 arithmetic progressions, $a \pmod{11184810}$, do not contain numbers of the form $p + 2^k$, where $a$ can be taken as the numbers in the last column of Tabel aptable. Additionally, we have identified other arithmetic progressions with moduli different from $11184810$ that do not contain numbe

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Definition
  • Theorem 4: Bang
  • Corollary 1
  • proof
  • Proposition 5
  • ...and 4 more