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

Abstract

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

10 pages