The infinitude of square-free palindromes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911386335969280 |
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| author | Johnston, Daniel R. Kerr, Bryce |
| author_facet | Johnston, Daniel R. Kerr, Bryce |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07097 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The infinitude of square-free palindromes Johnston, Daniel R. Kerr, Bryce Number Theory 11A63, 11N25 (Primary) 11L07 (Secondary) 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. |
| title | The infinitude of square-free palindromes |
| topic | Number Theory 11A63, 11N25 (Primary) 11L07 (Secondary) |
| url | https://arxiv.org/abs/2601.07097 |