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Bounded gaps and perfect power gaps in sequences of consecutive primes

Katalin Gyarmati

TL;DR

This paper investigates the fine structure of consecutive prime gaps, asking whether multiple successive gaps can all be large or equal to squares or perfect powers. It develops a synthesis of sieve methods, admissible-set techniques, and combinatorial density arguments to obtain quantitative bounds: a positive lower bound for $G_k(x)$, existence of square-difference prime pairs beyond any fixed threshold, and infinite runs of consecutive primes whose pairwise gaps are perfect powers. It also derives density-type bounds for square-difference-free prime sets and for prime subsets avoiding square differences via residue-class constructions and colorings, with extensive numerical verification in the Appendix. The results provide explicit, though very large, constants and demonstrate new ways to control gaps and difference-structure within prime sequences, highlighting a blend of analytic and combinatorial approaches in prime-gap problems.

Abstract

We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also presented.

Bounded gaps and perfect power gaps in sequences of consecutive primes

TL;DR

This paper investigates the fine structure of consecutive prime gaps, asking whether multiple successive gaps can all be large or equal to squares or perfect powers. It develops a synthesis of sieve methods, admissible-set techniques, and combinatorial density arguments to obtain quantitative bounds: a positive lower bound for , existence of square-difference prime pairs beyond any fixed threshold, and infinite runs of consecutive primes whose pairwise gaps are perfect powers. It also derives density-type bounds for square-difference-free prime sets and for prime subsets avoiding square differences via residue-class constructions and colorings, with extensive numerical verification in the Appendix. The results provide explicit, though very large, constants and demonstrate new ways to control gaps and difference-structure within prime sequences, highlighting a blend of analytic and combinatorial approaches in prime-gap problems.

Abstract

We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also presented.
Paper Structure (7 sections, 14 theorems, 100 equations)

This paper contains 7 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm01']}.
  3. Proof of Theorem \ref{['thm02']}.
  4. Proofs of Theorems \ref{['thm03']} and \ref{['thm04']}.
  5. Proof of Theorem \ref{['thm05']}.
  6. Proof of Theorem \ref{['thm06']}.
  7. Appendix

Key Result

Theorem 1

There exists a positive constant $x_0$ such that if $x_0<x\in\mathbb R$ and $2\le k\in\mathbb N$, then

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • Lemma 2: Halberstam
  • Lemma 3: Bloom-Maynard
  • Definition 1
  • ...and 7 more