Are there infinitely many palindromic primes to every base

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer base $b\ge2$, write the unique base-$b$ expansion of $N\ge1$ as $N=\sum_{j=0}^m a_jb^j$, where $m\ge0$, $0\le a_j<b$ and $a_m\ne0$. It is palindromic in base $b$ if $a_j=a_{m-j}$ for every $0\le j\le m$. A base-$b$ palindromic prime is a prime with this digit symmetry.

Statement recorded in the supplied source.

For every integer $b\ge2$, are there infinitely many primes palindromic in base $b$? In particular, is this true for base $10$? [S2]

Short English statement

Does each integer base have infinitely many primes whose digits read the same in both directions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400155 (AMR-093-0155). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many palindromic primes to every base. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal Sum of Palindromes, arXiv:1803.00161v2 (2018) Abstract and Section 1, p. 1. https://arxiv.org/pdf/1803.00161.