Powerful numbers in short intervals

Jean Marie De Koninck*, Florian Luca, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

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+0(1). We also show that the ABC-conjecture implies that for any fixed δ > 0 and sufficiently large N, the interval (N, N + N 1-(2+δ/κ) contains at most one κ-full number.

Original languageEnglish
Pages (from-to)11-16
Number of pages6
JournalBulletin of the Australian Mathematical Society
Volume71
Issue number1
Publication statusPublished - Feb 2005

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