Finiteness results for Diophantine triples with repdigit values

Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb N$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples $(a,b,c)\in \mathbb N^3$ with $c<b<a$ such that $ab+1,ac+1$ and $ab+1$ lie in the set $\mathcal R_g$. In this paper, we prove effective finitness results for the set $\mathcal D_g$

On XX-coordinates of Pell equations which are repdigits

Let $b\ge 2$ be a given integer. In this paper, we show that there only finitely many positive integers $d$ which are not squares, such that the Pell equation $X^2-dY^2=1$ has two positive integer solutions $(X,Y)$ with the property that their $X$-coordinates are base $b$-repdigits. Recall that a base $b$-repdigit is a positive integer all whose digits have the same value when written in base $b$. We also give an upper bound on the largest such $d$ in terms of $b$.Comment: To appear in The Fibonacci Quarterly Journa

On repdigits as product of consecutive Fibonacci numbers

Let (F$_{n}$)$_{n\geq0}$ be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn · · · F$_{n+(k-1)}$ is a repdigit, with at least two digits, then (k, n) = (1, 10)