Digitally delicate primes

TL;DR

This paper proves that a positive proportion of primes are so fragile that changing one digit or inserting digits at the ends makes them composite, extending Tao's earlier results on digital delicacy.

Contribution

The paper strengthens Tao's result by showing a positive proportion of primes become composite under any single-digit change or insertion of fixed digits at ends.

Findings

A positive proportion of primes are digitally delicate.

Changing one digit can make many primes composite.

Inserting fixed digits at ends also renders many primes composite.

Abstract

Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". We strengthen this result in a manner suggested by Tao: A positive proportion of primes become composite under any change of a single digit and any insertion a fixed number of arbitrary digits at the beginning or end.