Primes with one excluded digit
Volume 202 / 2022
Abstract
Given a base $b\geq 2$ and $a_0\in \{0,\ldots ,b-1\}$, an interesting number-theoretic question is whether there are infinitely many prime numbers having no digit $a_0$ in their representation in base $b$. Maynard (2019) answered this question affirmatively in the case $b=10$ and for sufficiently large $b$; the lower bound for $b$ which follows from Maynard’s proof is approximately $2\cdot 10^6$. We extend Maynard’s results by giving a proof for all $b\geq 250$ and reducing the verification of the claim for each $10\leq b\leq 249$ to Mathematica codes. The crucial step of this improvement is the estimation of some eigenvalues of matrices via the $\|\,\|_{\infty }$-norm. This finally gives an upper and a lower bound for the number of primes less than $X=b^k$ avoiding $a_0$ in their base $b$ representation. Hence, there are infinitely many primes of this form.
