Development of sieve of Eratosthenes and sieve of Sundaram's proof

TL;DR

This paper develops two algorithms based on sieves for generating primes up to a limit, improving on existing methods by using specific formulas and proving the sieve of Sundaram's algorithm, with performance comparisons.

Contribution

It introduces two new prime generation algorithms derived from sieves, proving and enhancing the sieve of Sundaram, and compares their efficiency with the classical sieve of Eratosthenes.

Findings

The algorithms successfully generate all primes up to a limit.

The improved sieve of Sundaram shows better efficiency than the original.

Performance analysis demonstrates the advantages over traditional sieve of Eratosthenes.

Abstract

We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm in terms of run time.