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Generating Prime Numbers -- A Fast New Method

V. Vilfred Kamalappan

Abstract

Bertrand's Postulate ensures existence of prime $p$ between $n$ and $2n$, $n$ an integer $\geq 2$ and the sieve of Eratosthenes, a very simple ancient algorithm, generates all prime numbers up to any given limit. Combining the above two, in this paper, we provide a simple fast moving algorithm to generate prime numbers up to any given limit. We also discuss Riemann zeta function related to generating of prime numbers.

Generating Prime Numbers -- A Fast New Method

Abstract

Bertrand's Postulate ensures existence of prime between and , an integer and the sieve of Eratosthenes, a very simple ancient algorithm, generates all prime numbers up to any given limit. Combining the above two, in this paper, we provide a simple fast moving algorithm to generate prime numbers up to any given limit. We also discuss Riemann zeta function related to generating of prime numbers.

Paper Structure

This paper contains 5 sections, 10 theorems, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result
  4. Algorithm to generate primes
  5. Riemann zeta function and prime numbers generation

Key Result

Lemma \oldthetheorem

Let $1 \leq i < j$, $i,j\in \mathbb{N}$, $p_i, p_j$ be primes, $Q_{i,j}$ = $\left\lfloor \frac{p_j}{p_i} \right\rfloor$, $Q_{i,j}^{'}$ = $\left\lfloor \frac{2p_j}{p_i}\right\rfloor$, $Q_{i,j}^{"}$ = $\left\lfloor \frac{p_j^2}{p_i} \right\rfloor$ and $Q_{i,j}^{"'}$ = $\left\lfloor \frac{p_j^2+4p_j+3}

Theorems & Definitions (31)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 21 more