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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