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14-term arithmetic progressions on quartic elliptic curves.

Journal of Integer Sequences [electronic only]

3 as a Ninth Power (mod p).

Mathematica Scandinavica

6 nombres dont les sommes deux à deux sont des carrés

Mémoires de la Société Mathématique de France

A Bogomolov property for curves modulo algebraic subgroups

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 Bound for Prime Solutions of Some Ternary Equations.

Mathematische Zeitschrift

Integers

Acta Arithmetica

A “class group” obstruction for the equation $C{y}^{d}=F\left(x,z\right)$

Journal de Théorie des Nombres de Bordeaux

In this paper, we study equations of the form $C{y}^{d}=F\left(x,z\right)$, where $F\in ℤ\left[x,z\right]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...

Acta Arithmetica

A class of diophantine equations connected with certain elliptic curves over $Q\left(\sqrt{-13}\right)$

Compositio Mathematica

A Classical Diophantine Problem and Modular Forms of Weight 3/2.

Inventiones mathematicae

Integers

Acta Arithmetica

A consequence of an effective form of the abc-conjecture

Colloquium Mathematicae

T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A ${A}^{0.4}$ with two exceptions given in Table 2.

A Diophantine equation in virus structure.

Mathematica Scandinavica

A diophantine system.

International Journal of Mathematics and Mathematical Sciences

Acta Arithmetica

Acta Arithmetica

A generalization of the question of Sierpiński on geometric progressions.

Journal of Integer Sequences [electronic only]

A “Hardy-Littlewood” approach to the $S$-unit equation

Compositio Mathematica

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