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Legendre Conjecture over Arithmetic Progressions

N. A. Carella

Abstract

Let $1\leq a<q$ be a pair of small integers such that $\gcd(a,q)=1$ and let $x>1$ be a large number. This note discusses the existence of a short sequence of primes $p\equiv a\bmod q$ between two squares $x^2$ and $(x+1)^2$.

Legendre Conjecture over Arithmetic Progressions

Abstract

Let be a pair of small integers such that and let be a large number. This note discusses the existence of a short sequence of primes between two squares and .
Paper Structure (4 sections, 6 theorems, 20 equations)

This paper contains 4 sections, 6 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Primes in Arithmetic Progressions
  3. Legendre Conjecture in Arithmetic Progressions
  4. Primes Between Two Cubic Integers and Beyond

Key Result

Theorem 2.1

Let $a$ and $q$ be integers with $1\leq q \leq 10^5$ and $\gcd(a, q) = 1$. If $x \geq 10^3$, then

Theorems & Definitions (12)

  • Theorem 2.1
  • proof : Proof
  • Lemma 2.1
  • proof : Proof
  • Theorem 3.1
  • proof : Proof
  • Lemma 3.1
  • proof : Proof
  • Theorem 3.2
  • proof : Proof
  • ...and 2 more