The infinitude of square-free palindromes

TL;DR

This paper proves that for any base greater than or equal to 2, there are infinitely many square-free palindromic numbers and provides an asymptotic count for them, using advanced number theory techniques.

Contribution

It establishes the infinitude of square-free palindromes in any base and derives an asymptotic formula for their distribution.

Findings

Infinitely many square-free palindromes exist in all bases ≥ 2.

An asymptotic expression for the count of such palindromes up to x.

Application of a hybrid p-adic/Archimedean van der Corput process.

Abstract

We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski.