Paper 2013/843

A Study of Goldbach's conjecture and Polignac's conjecture equivalence issues

Jian Ye and Chenglian Liu

Abstract

The famous Goldbach's conjecture and Polignac's conjecture are two of all unsolved problems in the field of number theory today. As well known, the Goldbach's conjecture and the Polignac's conjecture are equivalent. Most of the literatures does not introduce about internal equivalence in Polignac's conjecture. In this paper, we would like to discuss the internal equivalence to the Polignac's conjecture, say $T_{2k}(x)$ and $T(x)$ are equivalent. Since $T_{2k}\sim T(x)\sim 2c\cdot \frac{x}{(\ln x{)}^2}$, we rewrite and re-express to $T(x)\sim T_4(x)\sim T_8(x)\sim T_{16}(x)\sim T_{32}(x)\sim T_{2^n}(x)\sim 2c\cdot \frac{x}{(\ln x{)}^2}$. And then connected with the Goldbach's conjecture. Finally, we will point out the important prime number symmetry role of play in these two conjectures.