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Consecutive integers with no large prime factors

Published online by Cambridge University Press:  09 April 2009

R. B. Eggleton  and
J. L. Selfridge
J. L. Selfridge
Affiliation:
Department of Mathematical SciencesNorthern Illinois UniversityDeKalb, Illinois 60115, U.S.A.
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For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 1 , August 1976 , pp. 1 - 11
Copyright
Copyright © Australian Mathematical Society 1976

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