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On strings of consecutive integers with no large prime factors

Part of: Multiplicative number theory Sequences and sets

Published online by Cambridge University Press:  09 April 2009

Antal Balog  and
Trevor D. Wooley
Antal Balog
Affiliation:
Mathematical Institute Budapest1364Hungary e-mail: balog@math-inst.hu
Trevor D. Wooley
Affiliation:
Department of Mathematics University of Michigan East Hall525 East University Avenue Ann Arbor, Michigan 48109-1109USA e-mail: wooley@math.lsa.umich.edu
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Abstract

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We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

Keywords

Smooth numbersconsecutive integerssequences.

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints 11B25: Arithmetic progressions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 2 , April 1998 , pp. 266 - 276
Copyright
Copyright © Australian Mathematical Society 1998

References

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