Dekoninck 2005 powerful numbers short intervals

../

theorem_1: For every integer kappa >= 2 there are infinitely many N for which the open interval (N^kappa, (N+1)^kappa) contains at least ((3/8 + o(1)) log N / log log N)^(1/3) kappa-full integers.

theorem_2: The ABC conjecture implies that for fixed kappa and delta > 0 there is L_0 such that for L > L_0 the interval (L, L + L^(1-(2+delta)/kappa)) contains at most one kappa-full number.

theorem_3: For any positive integers L and K the interval (L, L+K) contains at most O(K log log K / log K) squarefull numbers.

---

De Koninck, Jean-Marie and Luca, Florian and Shparlinski, Igor E., Powerful numbers in short intervals. Bull. Austral. Math. Soc. 71 (2005), 11--16.

For an integer kappa >= 2, Theorem 1 shows there are infinitely many N such that the open interval (N^kappa,(N+1)^kappa) contains at least ((3/8+o(1)) log N / log log N)^{1/3} kappa-full integers, extending the squarefull case of De Koninck-Luca to all kappa. The proof replaces the continued-fraction argument by Roth's theorem (or the fully effective Liouville theorem for a slightly weaker uniform statement) combined with Dirichlet's simultaneous approximation theorem applied to alpha_j = d_j^{-1/kappa} for the first 2l squarefree numbers d_j greater than 1; this also gives a slightly better constant than the earlier paper. In the other direction Theorem 2 shows the ABC conjecture implies that for fixed kappa and delta>0 and large N the interval (N,N+N^{1-(2+delta)/kappa}) contains at most one kappa-full number, and Theorem 3 gives the unconditional but much weaker bound O(K log log K/log K) for the number of squarefull integers in an interval (L,L+K), hence for those that are kappa-full for some kappa >= 2. For problem 942 on powerful numbers between consecutive squares, Theorem 1 with kappa = 2 gives at least ((3/8+o(1)) log n / log log n)^{1/3} of them in (n^2,(n+1)^2) for infinitely many n. Theorem 2 is empty when kappa = 2, since the interval then has length below 1, and Theorem 3 with K about 2n gives only the upper bound O(n log log n/log n).

Source: https://doi.org/10.1017/S0004972700037953. No copyright line is printed; the copy read carries the journal's permissions notice "Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/05" on printed p. 11 and the footer "Published online by Cambridge University Press" on every page, and names no license, every other right reserved.

Bears on. #942: Theorem 1 with kappa = 2 gives at least ((3/8+o(1)) log n / log log n)^{1/3} powerful integers in (n^2,(n+1)^2), hence in [n^2,(n+1)^2), for infinitely many n, a lower bound along a sequence only; Theorem 3 with L = n^2 and K = 2n+1 gives at most 1 + O(n log log n / log n) of them for every n, far above the (log n)^{c+o(1)} the problem asks about; Theorem 2 says nothing when kappa = 2. None of these settles the problem, and the paper does not mention it.

Read status. Claims checked: the statements of Theorems 1, 2 and 3 were read clause by clause on the printed pages (pp. 11-16); the proofs were read for structure only.

Results.

• Theorem 1 (p. 12): for each integer kappa >= 2 there are infinitely many N with at least ((3/8+o(1)) log N / log log N)^{1/3} kappa-full integers in (N^kappa,(N+1)^kappa).
• Theorem 2 (p. 14): the ABC conjecture implies that if kappa and delta > 0 are fixed, there is L_0 such that for L > L_0 the interval (L,L+L^{1-(2+delta)/kappa}) contains at most one kappa-full number.
• Theorem 3 (p. 15): for any positive integers L and K the interval (L,L+K) contains at most O(K log log K / log K) squarefull numbers.

Not given a page: Conjecture 1 (p. 14), the ABC conjecture as the paper states it, which Theorem 2's page restates; and the closing remarks (pp. 15-16) on a version of Theorem 1 uniform in kappa via Liouville's theorem, recorded on Theorem 1's page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.