A remark on primality testing and decimal expansions

Details A remark on primality testing and decimal expansions A remark on primality testing and decimal expansions

2008-02-22

arxiv.org/abs/0802.3361v4

Mathematics

Number Theory

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

Scientific paper

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.

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