Proofs of the Twin Primes and Goldbach Conjectures

TL;DR

This paper introduces modular signatures based on prime residues, uses them within primorials, and employs elementary sieve and combinatorial methods to prove the twin primes and Goldbach conjectures.

Contribution

It presents a novel approach using modular signatures and elementary sieve properties to establish proofs for two longstanding conjectures.

Findings

Proof of the twin primes conjecture

Proof of the Goldbach conjecture

Introduction of modular signatures as a proof technique

Abstract

Associate a unique numerical sequence called the modular signature with each positive integer, using modular residues of each integer under the prime numbers, and distinguishing between the core seed primes and non-core seed primes used to create the modular signatures. Group the modular signatures within primorials. Use elementary sieve properties and combinatorial principles to prove the twin primes and Goldbach conjectures.