Power-free palindromes and reversed primes

TL;DR

This work studies the digital reverse in base $b$ and proves new results on weakened conjectures for reversed primes and palindromes. It establishes that for $b\ge 26000$ there are infinitely many primes $p$ with $\overleftarrow{p}$ square-free, and for all $b\ge 2$ there exist infinitely many cube-free palindromes, providing explicit asymptotic counting formulas. The proofs combine Telhcirid-style estimates for reversed primes in arithmetic progressions with equidistribution results for palindromes and Brun–Titchmarsh-type bounds, yielding asymptotics that detach the power-free condition from primality or palindromicity after local restrictions. The paper also discusses possible improvements to extend base ranges toward full conjectures and outlines the computational refinements essential to the base-range extension, illustrating a path forward for stronger results on square-free palindromes and reversed primes.

Abstract

We prove new results related to the digital reverse $\overleftarrow{n}$ of a positive integer $n$ in a fixed base $b$. First we show that for $b\geq 26000$, there exists infinitely many primes $p$ such that $\overleftarrow{p}$ is square-free. Further, we show that for $b\geq 2$ there are infinitely many palindromes (with $n=\overleftarrow{n}$) that are cube-free. We also give asymptotic expressions for the counting functions corresponding to these results. The main tools we use are recent bounds from the literature on reversed primes and palindromes in arithmetic progressions.

Figures (1)

• Figure 1: A plot of $f_h(\theta)$ generated by Desmos Desmos. Here, $b=3$ and $h=0$. Increasing $b$ increases the height of the peaks and reduces $L_b$. Changing $h$ shifts the plot along the $\theta$-axis.

Theorems & Definitions (22)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['revthm']}