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Powerful numbers in short intervals
Published online by Cambridge University Press: 17 April 2009
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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.
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- Copyright © Australian Mathematical Society 2005
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