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arXiv · 1806.11079

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

Patrick Morton
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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.

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Patrick Morton. 2018-06-28. Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, II: The Rogers-Ramanujan continued fraction. https://arxiv.org/abs/1806.11079

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