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Let m be a positive integer and let E be an elliptic curve over Q with the property that m | #E(F p ) for a density 1 set of primes p. Building upon work of Katz and Harron-Snowden, we study the probability that m | #E(Q) tor : we find it is nonzero for all m ∈ {1, 2, . . . , 10} ∪ {12, 16} and we compute it exactly when m ∈ {1, 2, 3, 4, 5, 7}. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve is torsion free of genus zero.

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The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are analogous to the Fermat numbers. We demonstrate that these sequences have many of the same properties as the Fermat numbers, and we discuss results about the prime factors of sequences generated by specific curves and points.