On sums of consecutive squares
Journal of Number Theory , Volume 62 - Issue 1 p. 39- 70
In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in thus sum is k2. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstraß equation with parameter k. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms of k. We conjecture that this point indeed generates the free part of the Mordell-Weil group and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. For k in the range 1 ≤k≤100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameter k. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms - which is an essential ingredient of our approach - the rank is the dominant factor. Also the extra computational effort that is needed for some values of k in order to determine the rank unconditionally and construct a set of generators for the Mordell-Weil group deserves special attention, as there are some unusual features.