A Diophantine problem concerning polygonal numbers (Q2865140)

Let NEWLINE\[NEWLINE\mathrm{Pol}^m_x=\frac{x((m-2)x+4-m)}{2}NEWLINE\]NEWLINE be the polygonal numbers with integral parameters \(x\geq 1\) and \(m\geq 3\). In this work, the authors consider the equation NEWLINE\[NEWLINE\mathrm{Pol}^m_x=(\mathrm{Pol}^n_y)^k\tag{1}NEWLINE\]NEWLINE where \(x>1\), \(y>1\) and \(k\geq 2\) are unknown integers, \(m\geq 3\) and \(n\geq 3\) are fixed integers. Firstly, they prove that (1) has only finitely many solutions where \(x>1\), \(y>1\), \(k\geq 2\), \(m\neq 4\) and they show that \(c_1\) is an effectively computable constant depending on \(m\) and \(n\) can be found such that \(\max\{x,y,k\}<c_1\). Secondly, for \(m=3,5,6,8\) and \(20\) they find all the solutions of the equation NEWLINE\[NEWLINE\mathrm{Pol}^m_x=z^kNEWLINE\]NEWLINE where \(x>1\), \(z>1\) and \(k\geq 3\) are integers. And also they give all the solutions of the equation (1) where \(k=2\) and \(3\leq m,n\leq 12\), \(m\neq 4\).NEWLINENEWLINEIn the proofs, they use some powerful finiteness theorems on modern theory of Diophantine equations which were proved by \textit{A. Schinzel} and \textit{R. Tijdeman} [Acta Arith. 31, 199--204 (1976; Zbl 0339.10018)] and \textit{B. Brindza} [Acta Math. Hung. 44, 133--139 (1984; Zbl 0552.10009)] and an important result of \textit{M. A. Bennett} [Bull. Lond. Math. Soc. 36, No. 5, 683--694 (2004; Zbl 1152.11321)] about the product of two consecutive integers.