A counterexample to the Pellian equation conjecture of Mordell
Tom 215 / 2024
Streszczenie
Let be a squarefree integer, let \omega \in \big \{\sqrt {d},\frac {1+\sqrt {d}}{2}\big \} be such that \mathbb {Z}[\omega ] is the ring of algebraic integers of the real quadratic number field \mathbb {Q}(\sqrt {d}), let \varepsilon \gt 1 be the fundamental unit of \mathbb {Z}[\omega ] and let x and y be the unique nonnegative integers with \varepsilon =x+y\omega . In this note, we extend and study the list of known squarefree integers d\geq 2, for which y is divisible by d (cf. OEIS A135735). As a byproduct, we present a counterexample to a conjecture of L. J. Mordell.