A Smooth Selberg Sieve and Applications

Springer Proceedings in Mathematics & Statistics - Geometry, Algebra, Number Theory, and Their Information Technology Applications ◽

10.1007/978-3-319-97379-1_14 ◽

2018 ◽

pp. 291-315

Author(s):

M. Ram Murty ◽

Akshaa Vatwani

Keyword(s):

Selberg Sieve

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The Selberg sieve for sequences

Arithmetical Aspects of the Large Sieve Inequality ◽

10.1007/978-93-86279-40-8_14 ◽

2009 ◽

pp. 111-117

Author(s):

Olivier Ramaré

Keyword(s):

Selberg Sieve

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A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788712000043 ◽

2011 ◽

Vol 91 (3) ◽

pp. 405-413 ◽

Cited By ~ 7

Author(s):

TERENCE TAO

Keyword(s):

Upper Bound ◽

Primality Testing ◽

Positive Proportion ◽

Fixed Base ◽

Covering Set ◽

Selberg Sieve

AbstractWe 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.

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