J. Korean Math. Soc. 2015; 52(4): 765-780
Printed July 1, 2015
https://doi.org/10.4134/JKMS.2015.52.4.765
Copyright © The Korean Mathematical Society.
Bojan Ba\v si\'c
University of Novi Sad
We consider the problem of characterizing the palindromic sequences $\langle c_{d-1},c_{d-2},\dots,c_0\rangle$, $c_{d-1}\neq 0$, having the property that for any $K\in\mathbb{N}$ there exists a number that is a palindrome simultaneously in $K$ different bases, with $\langle c_{d-1},c_{d-2},\dots,c_0\rangle$ being its digit sequence in one of those bases. Since each number is trivially a palindrome in all bases greater than itself, we impose the restriction that only palindromes with at least two digits are taken into account. We further consider a related problem, where we count only palindromes with a fixed number of digits (that is, $d$). The first problem turns out not to be very hard; we show that all the palindromic sequences have the required property, even with the additional point that we can actually restrict the counted palindromes to have at least $d$ digits. The second one is quite tougher; we show that all the palindromic sequences of length $d=3$ have the required property (and the same holds for $d=2$, based on some earlier results), while for larger values of $d$ we present some arguments showing that this tendency is quite likely to change.
Keywords: palindrome, number base, heuristic
MSC numbers: 11A63
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