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.
|Number of pages||6|
|Journal||Bulletin of the Australian Mathematical Society|
|Publication status||Published - Feb 2005|