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A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$ . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

A note on the number of $S$ -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

Let $(a_{1}, \dots, a_{m})$ be an $m$ -tuple of positive, pairwise distinct integers. If for all $1 \leq i < j \leq m$ the prime divisors of $a_{i} a_{j} + 1$ come from the same fixed set $S$ , then we call the $m$ -tuple $S$ -Diophantine. In this note we estimate the number of $S$ -Diophantine quadruples in terms of $| S | = r$ .

Displaying 1 – 8 of 8

A Bogomolov property for curves modulo algebraic subgroups

A note on the number of $S$ -Diophantine quadruples

A rationality condition for the existence of odd perfect numbers.

An additive divisor problem.

An Estimate for the Fundamental Solutions of a Generalized Pell Equation.

An irregular D(4)-quadruple cannot be extended to a quintuple

An upper bound for the G.C.D. of two linear recurring sequences

Asymptotic aspects of the Diophantine equation $p^{k} x^{n k} - z^{k} = l$

Currently displaying 1 – 8 of 8