A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

摘要

We show that for any fixed base α, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where α = 2 has already been established by Cohen and Selfridge [Not every number is the sum or difference of two prime powers’, Math. Comput. 29 (1975), 79–81] and Sun [‘On integers not of the form ±p~a ± q~b’, Proc. Amer. Math. Soc. 128 (2000), 997–1002], using some covering congruence ideas of Erdos. 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.