On equal values of pyramidal and polygonal numbers (Q1279778)

Let \(\text{Pyr}(y,n)\) be the \(y\)th pyramidal number of order \(n\): \((y+1)((n-2) y^2+(5-n)y)/6\), and let \(\text{Pol} (x,m)\) be the \(x\)th polygonal number of order \(m\): \([(m-2)x^2-(m-4)x]/2\). Consider \(\text{Pyr} (y,n)=\text{Pol}(x,m)\) in positive integers \(x,y\). Then the following theorem is proved: Let \(m\) and \(n\) be rational integers greater than 2. Apart from some effectively determinable exceptional pairs \((m,n)\), all the solutions of the previous equation satisfy \(\max(x,y)<C\), where \(C\) is an effectively computable constant depending only on \(m\) and \(n\).