Hostname: page-component-7dd5485656-k8lnt Total loading time: 0 Render date: 2025-10-28T00:07:22.828Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE SQUARE-FREE PARTS OF ⌊en!⌋

Published online by Cambridge University Press:  09 August 2007

FLORIAN LUCA  and
IGOR E. SHPARLINSKI
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México C.P. 58089, 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

Keywords

11B8311D4111N36

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 391 - 403
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Balasurya, S., Luca, F. and Shparlinski, I. E., Prime divisors of some recurrence sequence, Per. Math. Hungarica, to appear.Google Scholar
2. Bennett, M. A., Rational approximations to algebraic numbers of small height: the Diophantine equation |ax n −by n |=1, J. Reine Angew. Math. 535 (2001), 1–49.CrossRefGoogle Scholar
3. Everest, G., Poorten, A. van der, Shparlinski, I. and Ward, T., Recurrence sequences, Mathematical Surveys and Monographs 104 (Amer. Math. Soc., Providence, RI, 2003).CrossRefGoogle Scholar
4. Evertse, J.-H., Schlickewei, H. P. and Schmidt, W. M., Linear equations in variables which lie in a multiplicative group, Ann. Math. 155 (2002), 807–836.CrossRefGoogle Scholar
5. Garaev, M. Z., Luca, F. and Shparlinski, I. E., Catalan and Apéry numbers in residue classes, J. Combin. Theory Ser. A 113 (2006), 851–865.CrossRefGoogle Scholar
6. Luca, F., Prime divisors of binary holonomic sequences, Advances in App. Math., to appear.Google Scholar
7. Nagell, T., On a special class of Diophantine equations of the second degree, Ark. Mat. 3 (1954), 51–65.CrossRefGoogle Scholar
8. Tenenbaum, G., Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995).Google Scholar