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TT2011

Tao, Terence, "A remark on primality testing and decimal expansions," J. Aust. Math. Soc., 91:3 (2011) 405--413.  (http://dx.doi.org/10.1017/S1446788712000043) MR 2900615

Abstract: We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=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 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.