A remark on primality testing and decimal expansions.(English)Zbl 1251.11089

Following ideas of F. Cohen and J. L. Selfridge (for the base $$a=2$$) [Math. Comput. 29, 79–81 (1975; Zbl 0296.10029)] the present paper shows that, for any integer positive base $$a$$, there exist primes $$p$$ such that changing any of the digits in the base $$a$$ expansion of $$p$$ the resulting number is composite. As a consequence a deterministic primality test need to read all the digits in the base $$a$$ expansion of a positive integer candidate to decide if $$a$$ is prime or not.
More generally the paper proves the following result (Theorem 1.2): Let $$K$$ be a natural number. For all sufficiently large $$N$$ a positive proportion of primes in the interval $$[N, (1+1/K)N]$$ verifies that all the numbers $$|kp\pm ja^i|$$ are composite ($$a,j,k\in [1,K],\,\, i\in [1,K\log K])$$.
Section 1 introduces the problem and Section 2 gives the proof of Theorem 1.2. Instead of using a fully covering set of congruences (as Cohen and Selfridge) this proof take a partially covering using some special primes and then applies upper bound sieves. Finally Section 3 discusses possible variants and generalizations of Theorem 1.2.

MSC:

 11Y11 Primality 11N36 Applications of sieve methods 11A07 Congruences; primitive roots; residue systems

Zbl 0296.10029
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References:

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