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 structura

This is the accompanying code to the our paper "On the Prime Pairs". Each section pertains to a specific equation or operation. This serves as a way to illustrate the function of each part of the proof. Link to colab On the Prime Pairs By Suqin Ao and Lucy Lu Abstract By using a new method with Sieves, we can find all kinds of prime pairs including the twin primes, and prove that for any natural number k , there are infinitely many pairs of primes that differ by 2k . Also, b

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) a

The P versus NP problem was a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly...

The First 100,000 Twin Primes (First prime of each pair only) PDF Excel of first 1000 primes There are some neat patterns that arise from...