Integral points on elliptic curves y2 = x(x - 2m)(x + p)

We provide a description of the integral points on elliptic curves y2 = x(x-2m) x (x + p), where p and p + 2m are primes. In particular, we show that for m = 2 such a curve has no nontorsion integral point, and for m = 1 it has at most one such point (with y > 0). Our proofs rely upon numerical computations and a variety of results on quartic and other diophantine equations, combined with an elementary analysis.

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Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).