Journal of the Australian Mathematical Society

Research Article

A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

TERENCE TAOa1

a1 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA (email: tao@math.ucla.edu)

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 ±pa±qb’, 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.

(Received February 22 2008)

(Accepted July 04 2008)

2010 Mathematics subject classification

  • primary 35J10

Keywords and phrases

  • primality testing;
  • sieve theory;
  • covering congruences

Footnotes

The author is supported by NSF grant CCF-0649473 and a grant from the MacArthur Foundation.

Communicated by I. E. Shparlinski