An elliptic surface related to sums of consecutive squares (Q1329674)

The rational solutions \((x,y)\) to the equation \[ P(x+n-1)- P(x-1):= \sum_{j=0}^{n-1} (x+j)^ 2 =y^ 2 \qquad (n>1) \tag{1} \] can be described as rational points on a smooth quadric curve. As such a curve does admit a rational parametrization, all solutions of (1) can be explicitly given. The authors extend this problem by considering the rational elliptic curve \(P(n)- P(m)= y^ 2\) for \(n,m\in \mathbb{Q}\), which may be rewritten as \[ 6\eta^ 2= (x^ 3- x)- (t^ 3-t), \tag{2} \] where \(x=2n+1\), \(t=2m+1\), and \(\eta= 2y\). Equation (2) can be viewed as an elliptic curve \(E\) over the function field \(\mathbb{Q}(t)\) and it is shown that the group of \(\mathbb{Q}(t)\)-rational points \(E(\mathbb{Q} (t))\) is isomorphic to \(\mathbb{Z}\oplus \mathbb{Z}/ 2\mathbb{Z}\). Also explicit generators are given. Moreover, the minimal elliptic surface over the \(t\)-line, given by \[ y^ 2= (x^ 3- x)- (t^ 3-t), \tag{3} \] is shown to have rank 2. Again, generators are provided. The implications of these and related results to the original problem are discussed and illustrated by numerical examples.