A Geometric View of the Sieve of Eratosthenes

TL;DR

This paper explores the geometric structure underlying the Sieve of Eratosthenes, revealing symmetries and formulas that relate to prime distribution and the sieve's internal order.

Contribution

It introduces geometric concepts like Focals and Extremes, uncovering symmetries and providing formulas that deepen understanding of the sieve's structure.

Findings

Symmetry in the distribution of Focals

Existence of a geometric order in the sieve

A formula for the greatest remainder with the same quotient

Abstract

We study the geometry of the Sieve of Eratosthenes. We introduce some concepts as Focals and Extremes. We find a symmetry in the distribution of the Focals (all the information about the primes is contained into a small set of numbers). We find that there is a geometric order in the Sieve and we give a formula for the greatest remainder that returns the same quotient.