We prove that there are at most finitely many complex Î» = 0, 1 such that two points on the Legendre elliptic curve Y2 = X(X âˆ’ 1)(X âˆ’ Î») with coordinates X = 2, 3 both have finite order. This is aâ€¦ (More)

Abstract. Let a, b be given multiplicatively independent positive integers and let Ç« > 0. In a recent paper written jointly also with Y. Bugeaud we proved the upper bound exp(Ç«n) for gcd(a âˆ’ 1, b âˆ’â€¦ (More)

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalousâ€¦ (More)

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion pointsâ€¦ (More)

We prove that for integers a > b > c > 0, the greatest prime factor of (ab+1)(ac+1) tends to infinity with a. In particular, this settles a conjecture raised by GyÃ¶ry, Sarkozy and Stewart, predictingâ€¦ (More)

Let f, g be polynomials with complex coefficients. Then either f3 = g2, or deg(f3 âˆ’ g2) â‰¥ 2 deg f + 1. (Actually Davenport remarked that the method would yield a more general result, which he statedâ€¦ (More)

Generalizing a result of Pourchet, we show that, if Î±, Î² are power sums over Q satisfying suitable necessary assumptions, the length of the continued fraction for Î±(n)/Î²(n) tends to infinity as nâ†’âˆž.â€¦ (More)

Abstract. About fifty years ago Mahler [M] proved that if Î± > 1 is rational but not an integer and if 0 < l < 1, then the fractional part of Î± is > l apart from a finite set of integers n dependingâ€¦ (More)

Let g(x) be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if g(h(x)) has boundedly many terms, then h(x) âˆˆ C[x] must also have boundedly many terms. Solving an olderâ€¦ (More)

Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in Lâˆ— outside roots of unity. This lower bound depends only on the degree [L : K].