On the $X$-coordinates of Pell equations which are Tribonacci numbers

TL;DR

This paper investigates the rare occurrence of Tribonacci numbers as solutions' X-coordinates in Pell equations, establishing uniqueness results and characterizing exceptions.

Contribution

It provides a complete characterization of when Pell equation solutions' X-coordinates are Tribonacci numbers, including the identification of exceptional cases.

Findings

At most one Tribonacci number can be an X-coordinate in Pell solutions for non-square d

Complete characterization of exceptional cases where multiple solutions occur

Establishment of bounds on possible solutions involving Tribonacci numbers

Abstract

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely characterize.