Golden ratios, Lucas Sequences and the Quadratic Family

TL;DR

This paper investigates the relationship between generalized Fibonacci ratios, the Golden Ratio, and quadratic families, providing numerical evidence and a proof that these ratios belong to quadratic families, revealing deep connections in number theory.

Contribution

The paper introduces a proof that converging sequences of generalized Fibonacci ratios are contained within at least one quadratic family, linking Fibonacci ratios to quadratic number theory.

Findings

Generalized Fibonacci ratios converge to specific quadratic family members.

Numerical evidence shows overlap between Fibonacci ratios and quadratic families.

A proof confirms that these ratios belong to quadratic families.

Abstract

It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic depiction of those ratios show some overlap with the quadratic family, and some numerical evidence suggest that everyone of those ratios in the finite set obtained, belong to at least one quadratic family, and finally a proof is presented that the converging sequence of some generalized Fibonacci ratios belong to at least one quadratic family.