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For all totally positive algebraic numbers α except a finite number of explicit exceptions, the following inequality holds: 1 d (α 1 + · · · + α d) > max(1.780022, 1.66 + α 1), where d is the degree of α and 0 < α 1 < · · · < α d its conjugates. This improves previous results of Smyth, Flammang and Rhin.

We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic curve over Q(t) induced by this triple, i.e.: y 2 = (c 1 (t) x + 1)(c 2 (t) x + 1)(c 3 (t) x + 1) has torsion group isomorphic to Z/2Z × Z/2Z and rank 5. This represents an improvement of the result of A. Dujella, who showed a family of this kind with rank 4. By… (More)

We consider arithmetic progressions on Pellian equations x 2 − d y 2 = m, i.e. integral solutions such that the y-coordinates are in arithmetic progression. We construct explicit infinite families of d, m for which there exists a five-term arithmetic progression in the y-coordinate, and we prove the existence of infinitely many pairs d, m parametrized by… (More)

There are 26 possibilities for the torsion group of elliptic curves dened over quadratic number elds. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over… (More)

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