On the Prime Pairs - Paper
By Suqin Ao and Lucy Lu
Abstract
The distribution of prime numbers exhibits a tension between randomness and rigid arithmetic structure. In this
paper, we utilize a modified Sieve of Eratosthenes to explore the connection between two of the most famous problems
in number theory: the Twin Prime Conjecture and the Goldbach Conjecture.
By analyzing the sieving density, we demonstrate that the count of twin primes and the count of Goldbach partitions share a fundamental structural link. We derive a heuristic formula suggesting that the number of Goldbach partitions η(2n) grows in proportion to the density
of twin primes, denoted by η(2n) ∼ Hπ2(n). This framework offers a fresh perspective on why these configurations of
primes are not merely accidental, but statistically inevitable consequences of sieve theory.
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Arxiv version