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Powerful numbers in short intervals

Published online by Cambridge University Press:  17 April 2009

Jean-Marie De Koninck ,
Florian Luca  and
Igor E. Shparlinski
Jean-Marie De Koninck
Affiliation:
Départment de Mathématiques, Université Laval, Québec G1K 7P4, Canada, e-mail: jmdk@mat.ulaval.ca
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México, e-mail: fluca@matmor.unam.mx
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
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Let κ ≥ 2 be an integer. We show that there exist infinitely many positive integers N such that the number of κ-full integers in the interval (Nκ, (N + 1)κ) is at least (log N)1/3+ο(1). We also show that the ABC-conjecture implies that for any fixed δ > 0 and sufficiently large N, the interval (N, N + N1−(2+δ)/κ) contains at most one κ-full number.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 11 - 16
Copyright
Copyright © Australian Mathematical Society 2005

References

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