Fibonacci numbers and Ford circles

Chords of an ellipse, Lucas polynomials, and cubic equations

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. …

Horocyclic Orbits on $Γ(1)\frontslash\mathcal{H}$, \ Closed and Otherwise

This paper studies certain horocyclic orbits on $Γ(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles defined using the pencil of circles whose common point (in the words of the Nielsen-Fenchel manuscript is $\infty$. The orbits involved in this case are closed and long - judged by arc length between two points compared to the hyperbolic distance between them. We are concerned with tracking the paths of individual horocycles. Using Ford circles of Farey sequences we find lifts to the Standard Fundamental Region (SFR) and find points of these lifts making given angles with a…

Lucas序列中的周期模式、进入点与阶：理论与应用

Period patterns, entry points, and orders in the Lucas sequences: theory and applications

本文将模 m 的 Fibonacci 和 Lucas 序列统计理论扩展至 Lucas 序列 U 和 V，基于阶、周期与进入点等概念推广了统计恒等式，并应用于新颖的图解方法以揭示序列项的周期模式。