When is a polygonal pyramid number again polygonal?
Lucas' so-called ``cannonball problem'' asks for the values of \(n\in\mathbb{N}\) for which the sum of the first \(n\) perfect squares is itself a perfect square. The only nontrivial solution is known to be \(n= 24\). In the present paper the authors consider the following generalizations: when is the sum of the first \(m\)-gonal numbers once again an \(m\)-gonal number, or more generally, a polygonal number of possibly different order \(n\)? The \(i\)th \(m\)-gonal number is \(G_m(i)= {1\over 2} (m- 2)i^2- {1\over 2}(m- 4)i\), and the \(i\)th pyramid number of order \(m\) is \(\text{Pyr}_m(i)= \sum^i_{j=1}G_m(j)\). So, given integers \(m, n\geq 3\) the, problem asks for all pairs \((x, y)\) of positive integers for which \(G_n(y)= \text{Pyr}_m(x)\). This equation can be explicitly written as an equation for an elliptic curve \(E_{m,n}\) over \(\mathbb{Q}\) depending on the parameters \(m\) and \(n\). The authors are able to find quite a few (infinite) families of solutions, typical examples of which are \[ \begin{aligned} \text{Pyr}_m(3k- 1) &= G_{(m-2)k+3}(3k- 1),\\ G_n(8k^2- 6k+ 1)&= \text{Pyr}_{3(n-2)k+2}(4k- 2).\end{aligned} \] The authors also prove that, when \(m\) and \(n\) are both viewed as independent variables, exactly one \(\mathbb{Q}[m, n]\)-point exists on the elliptic curve \(E_{m,n}\). They work with standard techniques from the theory of elliptic curves over rational function fields.
scientific article; zbMATH DE number 2012030