# Unity formula that connect the fine-structure constant and the proton to electron mass ratio

11 Pages Posted: 18 Nov 2021 Last revised: 16 Jun 2023

Date Written: November 8, 2021

### Abstract

In this paper will be presented the unity formulas that connect the fine-structure constant and the proton to electron mass ratio. The equations are simple,elegant and symmetrical in a great physical meaning. At the beginning we will make a review of the last works. We will suggest the exact formula for fine-structure constant in terms of the golden angle,the relativity factor and the fifth power of the golden mean. Also we propose a simple and absolutely accurate expression for the fine-structure constant in terms of the Archimedes constant. After we propose two exact mathematical expressions for the proton to electron mass ratio. It will be presented the mathematical formulas that connects the proton to electron mass ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro's number NA,the gravitational coupling constant αG of the electron and the gravitational coupling constant of the proton αG(p). Ιt will be explained that the product of the proton to electron mass ratio with the reciprocal of the fine-structure constant is one of the roots of a trigonometric equation. It will also be shown another way to show this trigonometric equation with the exponential form of the equation. The beautiful formula that connects the fine-structure constant,the proton to electron mass ratio and the fifth power of the golden mean. Also the formula that connects the fine-structure constant,the proton to electron mass ratio and mathematical constants π,φ,e and i. Finally we will show that the speed of light and the maximum speed of sound depends only from the fine-structure constant and not from the proton to electron mass ratio.

**Keywords:** Fine-structure constant , Proton to electron mass ratio , Dimensionless physical constants , Maximum speed of sound , Speed of light , Trigonometric Functions , Exponential Function

**Suggested Citation:**
Suggested Citation