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A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  19 March 2012

TERENCE TAO
TERENCE TAO*
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA (email: [email protected])
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Abstract

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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.

Keywords

primality testingsieve theorycovering congruences

MSC classification

Secondary: 35J10: Schrödinger operator
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 3 , December 2011 , pp. 405 - 413
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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