Relations between e, $\pi$, golden ratios and $\sqrt{2}$

TL;DR

This paper explores mathematical relationships among fundamental constants and ratios such as e, pi, the golden ratio, and sqrt(2), through additive p-sequences extending Fibonacci numbers.

Contribution

It introduces and analyzes relations between key mathematical constants and ratios using additive p-sequences, extending Fibonacci concepts.

Findings

Identifies connections between e, pi, golden ratios, and sqrt(2)

Defines additive p-sequences as a generalization of Fibonacci numbers

Establishes mathematical relations linking these constants and sequences

Abstract

We write out relations between the base of natural logarithms ($e$), the ratio of the circumference of a circle to its diameter ($\pi$), the golden ratios ($\Phi_p$) of the additive $p$-sequences, and the ratio of the diagonal of a square to its side ($\sqrt{2}$). An additive $p$-sequence is a natural extension of the Fibonacci sequence in which every term is the sum of $p$-previous terms given $p \ge 1$ initial values called seeds.