We prove new results related to the digital reverse <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overleftarrow{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>n</mi> <mo stretchy="false">←</mo> </mover> </math></EquationSource> </InlineEquation> of a positive integer n in a fixed base b. First we show that for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b\ge 26000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>26000</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists infinitely many primes p such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overleftarrow{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>p</mi> <mo stretchy="false">←</mo> </mover> </math></EquationSource> </InlineEquation> is square-free. Further, we show that for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> there are infinitely many palindromes (with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n=\overleftarrow{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mover accent="true"> <mi>n</mi> <mo stretchy="false">←</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>) 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.

Power-free palindromes and reversed primes

期刊论文
发表日期：

2025年10月31日

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Shashi Chourasiya,
Daniel R. Johnston

摘要

We prove new results related to the digital reverse $\overleftarrow{n}$ $\overset{\leftarrow}{n}$ of a positive integer n in a fixed base b. First we show that for $b\ge 26000$ $b \geq 26000$ , there exists infinitely many primes p such that $\overleftarrow{p}$ $\overset{\leftarrow}{p}$ is square-free. Further, we show that for $b\ge 2$ $b \geq 2$ there are infinitely many palindromes (with $n=\overleftarrow{n}$ $n = \overset{\leftarrow}{n}$ ) 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.