Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, II: The Rogers-Ramanujan continued fraction

TL;DR

This paper demonstrates that solutions to a specific Diophantine equation are given by values of the Rogers-Ramanujan continued fraction, which are shown to be periodic points of an algebraic function within certain quadratic field extensions.

Contribution

It establishes a link between solutions of a Diophantine equation and periodic points of an algebraic function via the Rogers-Ramanujan continued fraction in quadratic fields.

Findings

Solutions lie in abelian extensions of quadratic fields where -d ≡ ±1 mod 5

Coordinates are values of the Rogers-Ramanujan continued fraction

These values are periodic points of an algebraic function

Abstract

In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(\tau)$, and are shown to be periodic points of an algebraic function.