Powerful numbers in short intervals

Jean Marie De Koninck; Florian Luca; Igor E. Shparlinski

Powerful numbers in short intervals

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

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

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 (N^κ, (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 language	English
Pages (from-to)	11-16
Number of pages	6
Journal	Bulletin of the Australian Mathematical Society
Volume	71
Issue number	1
Publication status	Published - Feb 2005

Cite this

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title = "Powerful numbers in short intervals",

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.",

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AU - De Koninck, Jean Marie

AU - Luca, Florian

AU - Shparlinski, Igor E.

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N2 - 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.

AB - 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.

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JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

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Powerful numbers in short intervals

Abstract

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