Some Notes on Digit Strings in the Primes

TL;DR

This paper establishes explicit bounds on primes containing a specific digit string and demonstrates, via Green-Tao theorem, the existence of arbitrarily long prime progressions with that property.

Contribution

It provides explicit upper bounds for primes with a given digit string and extends Green-Tao theorem results to primes containing that string.

Findings

Explicit upper bounds for primes containing a specific digit string

Existence of arbitrarily long prime arithmetic progressions with a given digit string

Extension of Green-Tao theorem to primes with digit constraints

Abstract

Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long arithmetic progressions of prime numbers all of whose decimal representations contain $S$.