We investigate power values of sums of products of consecutive integers. We give general finiteness results, and also give all solutions when the number of terms in the sum considered is at most ten.
Let be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are and , respectively. We show that the Diophantine equation has only finitely many solutions in , where , is even and . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on...
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