Reasoning about Primes (I)

TL;DR

This paper introduces a novel theoretical sieve using a 'hydra' data structure to prove longstanding conjectures about primes, including twin primes, Kronecker's, and Polignac's conjectures, through elementary recursive methods.

Contribution

The paper presents a new 'hydra' data structure and recursive splitting algorithm that enable elementary proofs of major prime conjectures previously unproven.

Findings

Proves the twin prime conjecture.

Establishes the generalized conjectures of Kronecker and Polignac.

Provides elementary recursive proofs for prime distribution conjectures.

Abstract

We prove the twin prime conjecture and the generalized conjectures of Kronecker and Polignac. Key to the proofs is a new theoretical sieve that combines two concepts that go back to Eratosthenes: the 'sieve' filtering a finite set of numbers and the 'hydra' as a representation of infinity. Using functional programming notation (and a reference implementation in R) we define a data structure 'hydra' that partitions the infinite set of numbers into a finite set of partitions. On top of this data structure we define an algorithm 'split' which sub-partitions each partition using the modulus prime function. Hydra splitting along the natural sequence of primes is a recursive version of wheel factorization. Hydra recursion allows elementary proofs of some statements about primes. We consider one new and two classical proof structures. Using these proof methods we find that hydra recursion proves the infinity of twin primes. Then we show how to use specific selections of primes to create hydras that contain pairs of partitions with arbitrary even distance, which proves Maillet's conjecture and the stricter 'consecutive existence conjecture'. Together with our proof methodology we obtain elementary proofs of Kronecker's and Polignac's conjecture.