On $d$-digit palindromes in different bases: The number of bases is unbounded

Abstract

The following problem was posed in [E. H. Goins, Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins, \emph{Integers} \textbf{9} (2009), 725--734]: ``What is the largest list of bases $b$ for which an integer $N\geqslant 10$ is a $d$-digit palindrome base $b$ for every base in the list?" We show that it is possible to construct such a list as large as we please. Furthermore, we show that it is possible to construct such arbitrarily large list for \emph{any} given $d$.