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 X5+Y5=ε5(1−X5Y5) , where ε=−1+5√2 , has solutions in specific abelian extensions of quadratic fields K=Q(−d−−−√) in which −d≡±1 (mod 5 ). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction r(τ) , and are shown to be periodic points of an algebraic function.