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

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LAL_arxiv_10_symmetryinsieve.pdf

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