Study of Prime Pairs - Arithmetic Mirror

By S. Ao and L. Lu

Abstract

Twin Primes represent the closest possible spacing of prime numbers. The Goldbach Conjecture represents the

distinct ways primes can combine. For centuries, these have been treated as separate mysteries of Number Theory.

In this paper, we demonstrate that these two problems are actually mirror images of the same underlying arithmetic

structure.

By visualizing the "Double Sieve" mechanism, we show that the subtraction logic of Twin Primes (p; p+2) and the addition logic of Goldbach Partitions (p;2n􀀀 p) generate statistically identical density patterns. We propose a heuristic "Resonance Factor" that explains the famous "Goldbach Comet" and suggests that the infinitude of Twin Primes and the validity of the Goldbach Conjecture are not accidental, but are inevitable consequences of the deterministic interference patterns of the Sieve of Eratosthenes.

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