A remark on primality testing and decimal expansions

TL;DR

This paper demonstrates that a positive proportion of primes become composite when any one digit in their base $a$ expansion is altered, highlighting limitations in prime testing based solely on digit inspection.

Contribution

It introduces a new method using partially covering congruences and the Selberg sieve to analyze digit-altering effects on primes, extending previous results for base 2.

Findings

A positive proportion of primes become composite after changing any one digit in their base $a$ expansion.

Prime testing from base $a$ digits alone may be unreliable due to these digit-altering properties.

The method generalizes previous results and offers new insights into prime digit structure.

Abstract

We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erd\H{o}s. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base $a$ expansion without reading all of its digits. We also present some slight generalisations of these results.