The Twin Primes Seen from a Different Perspective

Abstract

The paper presents a framework for the construction of an elementary proof of the infinitude of twin primes. It starts from the fact that all positive integers can be divided into numbers that can lead directly to a pair of twin primes (called twin ranks) and numbers (called non-ranks) that do not have this property. While the twin ranks cannot be directly calculated, the non-ranks can be easily calculated with a simple equation based on ordinary primes. They present a series of properties that once rigorously proven make the finiteness of twin prime an impossibility. Foremost among these properties is the fact that they can be arranged in an infinite number of sets called groups and super-groups. These sets have a built-in symmetry, a precise interval length and a well-defined number of terms. Another important property is that the depletion of twin primes via non-ranks goes in steps from one “basic” interval to another. As one goes higher up in the number series, these intervals grow larger and larger while the prime numbers required for their depletion become more and more sparse.