On sums of consecutive squares
The authors consider the problem of deciding when a sum of consecutive squares is itself a square. More precisely, they aim at determining all pairs of integers \((n,t)\) for which the relation \[ k^2+(k+1)^2+ \cdots +\bigl(k+(n-1)\bigr)^2 =t^2 \tag{*} \] holds for a fixed parameter \(k\in\mathbb{Z}\). This problem amounts to the task of computing all integral points on the elliptic curve \[ E_k:y^2= x^3-36x- 864k(k-1) (2k-1) \] over \(\mathbb{Q}\). The latter task is solved for \(k\in\mathbb{Z}\) in the range \(1\leq k\leq 100\) by applying a well-known algorithm developed by \textit{R. J. Stroeker} and \textit{N. Tzanakis} [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)] and, independently, by \textit{J. Gebel}, \textit{A. Pethö} and \textit{H. G. Zimmer} [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)]. The algorithm requires the knowledge of the rank and a basis of the Mordell-Weil group \(E_k(\mathbb{Q})\), which is determined by employing the APECS program and could, as the authors point out, also be determined by means of the SIMATH package. However, there are a few undecided cases, when one uses APECS, and there is one undecided case with SIMATH (left open also by APECS), viz. \(k=68\). In this exceptional case the authors use a descent argument. In fact, by working in the class group of a specific quartic number field of class number 16128, they are able to show that a certain quartic has no global solution. The curve \(E_k\) has torsion group \(E_k(\mathbb{Q})_{tors} \cong \mathbb{Z}/2 \mathbb{Z}\) for \(k\geq 2\) with torsion point \(T_k=(6(2k-1),0)\) of order 2 and the obvious rational point of infinite order \(Q_k= (12k+ 6,72k)\in E_k (\mathbb{Q})\). The authors then state the following remarkable Conjecture. If \(E_k\) has rank 1 over \(\mathbb{Q}\), then \(Q_k\) generates the Mordell-Weil group \(E_k(\mathbb{Q})\) modulo the torsion group \(E_k(\mathbb{Q})_{tors}\). The conjecture is shown to be true for the rank-one curves in the range \(1\leq k\leq 100\). By some tedious calculations involving the classical \(\psi\)-functions, the authors eventually prove that neither \(mQ_k\) \((m>1)\) nor \(mQ_k +T_k\) \((m\geq 1)\) can be an integer point. Hence, modulo the above conjecture, the only integer solution of (*) for \(k\geq 2\) is \((n,t)=(1,k)\).
scientific article; zbMATH DE number 992093