Integral Points in Families of Elliptic Curves

The number of integral points on any given elliptic curve is finite. Taking a family of elliptic curves and imposing some ordering, we expect that very few curves have non-trivial integral points. In certain quadratic and cubic twist families, we prove that almost all curves contain no nontrivial integral points. The proof uses a correspondence by Mordell between integral points on elliptic curves and integral binary quartic forms.

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