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

Abstract

In this part we show that the diophantine equation $X^5+Y^5={\epsilon }^5(1-X^5{Y}^5)$, where $\epsilon =\frac{-1+\sqrt{5}}{2}$, 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.

44 pages, 2 tables, Section 5 of the paper has been substantially revised and corrected