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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.

- Juli´an Aguirre, Fernando Casta˜neda, Juan Carlos Peral
- 2003

We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, all of them with torsion group Z/(2 Z) and modular invariant j = 1728. We use it to look for curves of high rank in this family and present four such curves of rank 13 and 22 of rank 12.

We construct an elliptic curve over the eld of rational functions with torsion group Z/2Z × Z/4Z and rank equal to 4, and an elliptic curve over Q with the same torsion group and rank 9. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

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)

We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z × Z/6Z. Also in both cases we prove the existence of infinitely many elliptic curves, which are parameterized by the points of an elliptic curve with positive rank, with such torsion group and rank at least 3. These results… (More)

Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E… (More)

Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v 2 = u(u − a)(u − b)(u − c). The Heron formula states that Q = √ P (P − a)(P − b)(P − c) where P is the semi-perimeter of the triangle, so the point (u, v) = (P, Q) is a rational point on the quartic. Also the point of… (More)

- Andrej Dujella, Juan Carlos Peral, Petra Tadi
- 2015

We exhibit several families of elliptic curves with torsion group isomorphic to Z/6Z and generic rank at least 3. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we… (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)