Multiplicative diophantine equations of a special type
Volume 156 / 2019
Colloquium Mathematicum 156 (2019), 287-294
MSC: Primary 11D57; Secondary 11A05.
DOI: 10.4064/cm7362-8-2018
Published online: 7 February 2019
Abstract
We investigate the diophantine equation $$ \prod _{j=1}^kz_j(n-z_j)=t^q $$ where $k$, $n$ are treated as parameters and unknowns $t,z_1,\ldots ,z_k$ fulfill the normalizing condition $$ 1\leq z_1 \lt \cdots \lt z_k\leq {n/2}. $$ For $q=2$ we prove that for $n\not \in \{1,3,7\}$ there exists $k\in \{1,2,4\}$ such that the above equation has a solution satisfying the above restriction. For general $q\geq 3$ we provide some partial results.
