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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 MéxicoC.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
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Abstract

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

References

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