Twins of powerful numbers (Q2637428)

A positive integer \(n\) is called \(k\)-full (\(k\geq 2\) an integer) if \(p^k | n\) whenever \(p\) is a prime factor of \(n\). The problem investigated in this paper is an upper bound for the count of the number of solutions of the equation \[ n-m=\ell \] with \(\ell\neq 0\) fixed and \(n,m\leq x\) both \(k\)-full. Let \(N_k(x,\ell)\) be the number of such pairs. One expects that \(N_2(x,\ell)\ll x^{\varepsilon}\) for any \(\varepsilon>0\) and that \(N_k(x,\ell)=O(1)\) for \(k\geq 3\) and this follows under the \(abc\)-conjecture. In the paper under review, the authors show that \(N_k(x,\ell)\ll_{\varepsilon.k} x^{2/(2k+1)+\varepsilon}\) valid for all \(k\geq 2\) and \(\varepsilon>0\), where the implied constant in the Vinogradov symbol depends on \(k\) and \(\varepsilon>0\) but not on \(\ell\). The argument is elementary based on the size of the divisor function in real quadratic fields (counts for solutions up to a certain height to Pellian and norm form equations in real quadratic fields). For \(2\leq k\leq 5\), the authors use a result of \textit{M. N. Huxley} [Bonn. Math. Schr. 360, 36 p. (2003; Zbl 1065.11052)] to improve upon the above general bound. For example, they obtain \(N_2(x,\ell)\ll x^{61/180}\), thus improving a recent result of \textit{T. H. Chan} [J. Aust. Math. Soc. 93, No. 1--2, 43--51 (2012; Zbl 1290.11011)], who showed that \(N_2(x,\ell)\ll x^{7/19}\log x\) (note that \(61/180=0.338\ldots<7/19=0.368\ldots\)).