**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.

**Category / Keywords: **Goldbach's conjecture; Polignac's conjecture; Equivalent;

**Date: **received 13 Dec 2013, last revised 19 Dec 2013

**Contact author: **chenglian liu at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20131220:013454 (All versions of this report)

**Short URL: **ia.cr/2013/843

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