A remark on primality testing and decimal expansions

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 Erdős. 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.

arXiv comment9 pages, no figures, to appear, J. Aust. Math. Soc. Some references added

References in corpus (1)

• On the integers not of the form $p+2^a+2^b$