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Source. Theorem 2, p. 14, of Jean-Marie De Koninck, Florian Luca and Igor E. Shparlinski, Powerful numbers in short intervals, Bull. Austral. Math. Soc. 71 (2005), 11--16, doi:10.1017/S0004972700037953. See the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof (pp. 14--15) was read for structure only. Nothing here is independently reviewed.

Statement

The paper's Conjecture 1 (p. 14) is the ABC conjecture in the form: for every $\varepsilon>0$ there is $C(\varepsilon)$ such that $\max\{|a|,|b|,|c|\}\le C(\varepsilon)\gamma(abc)^{1+\varepsilon}$ for all integers $a,b,c$ with $c=a+b$ and $\gcd(a,b)=1$, where $\gamma(m)$ is the product of the primes dividing $m$. $\kappa$-full is defined as on Theorem 1, with $\kappa>1$ an integer.

Theorem 2 (p. 14). Assume the ABC conjecture. If $\kappa$ and $\delta>0$ are fixed, there is $L_0$ such that for every $L>L_0$ the interval $(L,L+L^{1-(2+\delta)/\kappa})$ contains at most one $\kappa$-full number.

The paper remarks (p. 15) that the best known results towards ABC, those of Stewart and Yu, are too weak to give any nontrivial unconditional estimate of this kind.

Proof pointer

Pp. 14--15. Two $\kappa$-full numbers $a<b$ in the interval have radicals at most $(2L)^{1/\kappa}$ and a difference below $L^{1-(2+\delta)/\kappa}$; applying ABC with $\varepsilon=\delta/\kappa$ to $b-a$ bounds $b$, and hence $L$, by a constant depending on $\kappa$ and $\delta$ only.

Dependencies

The ABC conjecture (the paper's Conjecture 1, p. 14).

Bears on

• Problem 942: none for $\kappa=2$. The interval then has length $L^{-\delta/2}<1$, so the conclusion holds trivially and says nothing about powerful numbers between consecutive squares. The paper does not mention the problem.