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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: [email protected]
Trevor D. Wooley
Affiliation:
Department of Mathematics University of Michigan East Hall525 East University Avenue Ann Arbor, Michigan 48109-1109USA e-mail: [email protected]
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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

[1]Balog, A., Erdös, P. and Tenenbaum, G., ‘On arithmetic functions involving consecutive divisors’, in: Analytic number theory (Allerton Park, IL, 1989) Progr. Math. 85, (Birkhäuser, Boston, 1990) pp. 77–90.CrossRefGoogle Scholar
[2]Balog, A. and Ruzsa, I. Z., ‘On an additive property of stable sets’, in: Sieve methods, exponential sums and their applications in number theory, Cardiff, 1995, London Mathematical Society Lecture Notes No. 237 (Cambridge University Press, Cambridge, 1997), pp. 55–63.CrossRefGoogle Scholar
[3]Eggleton, R. B. and Selfridge, J. L., ‘Consecutive integers with no large prime factors’, J. Austral. Math. Soc. Ser. A 22 (1976), 1–11.CrossRefGoogle Scholar
[4]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 5th edition (Clarendon Press, Oxford, 1989).Google Scholar
[5]Heath-Brown, D. R., ‘Consecutive almost-primes’, J. Indian Math. Soc. 52 (1987), 39–49.Google Scholar
[6]Heath-Brown, D. R., ‘Correction and Footnote to “Consecutive almost-primes”’, (J. Indian Math. Soc. 52 (1987), 39–49), J. Indian Math. Soc., to appear.Google Scholar
[7]Hildebrand, A., ‘On a conjecture of Balog’, Proc. Amer. Math. Soc. 95 (1985), 517–523.CrossRefGoogle Scholar
[8]Hildebrand, A., ‘On integer sets containing strings of consecutive integers’, Mathematika 36 (1989), 60–70.CrossRefGoogle Scholar
[9]Hildebrand, A. and Tenenbaum, G., ‘Integers without large prime factors’, J. Théor. Nombres Bordeaux 5 (1993), 411–484.CrossRefGoogle Scholar
[10]Szemerédi, E., ‘On sets of integers containing no k terms in arithmetic progression’, Acta Arith. 27 (1975), 199–245.CrossRefGoogle Scholar