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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: [email protected]
Florian Luca
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58180, Morelia, Michoacán, México, e-mail: [email protected]
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: [email protected]
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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

[1]Bateman, P. and Grosswald, E., ‘On a theorem of Erdös and Szekeres’, Illinois J. Math. 2 (1958), 88–98.CrossRefGoogle Scholar
[2]Halberstam, H. and Richert, H.-E., Sieve methods (Academic Press, London, 1974).Google Scholar
[3]De Koninck, J.-M. and Luca, F., ‘Sur la proximité des nombres puissants’, Acta Arith. 114 (2004), 149–157.CrossRefGoogle Scholar
[4]Loxton, J.H., ‘Some problems involving powers of integers’, Acta Arith. 46 (1986), 113–123.CrossRefGoogle Scholar
[5]Schmidt, W.M., Diophantine approximation (Springer-Verlag, Berlin, 1980).Google Scholar
[6]Stewart, C.L. and Yu, K.R., ‘On the abc conjecture, II’, Duke Math. J. 108 (2001), 169–181.CrossRefGoogle Scholar