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Determinants of knots and Diophantine equations

A. Stoimenow (2007)

Acta Arithmetica

Diagonal equations of different degrees over p-adic fields

Michael P. Knapp (2007)

Acta Arithmetica

Diophantine equation $\frac{q^{n} - 1}{q - 1} = y$ for four prime divisors of $y - 1$

Zdeněk Polický (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper the special diophantine equation $\frac{q^{n} - 1}{q - 1} = y$ with integer coefficients is discussed and integer solutions are sought. This equation is solved completely just for four prime divisors of $y - 1$ .

Diophantine triples with values in binary recurrences

Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we study triples $a, b$ and $c$ of distinct positive integers such that $a b + 1, a c + 1$ and $b c + 1$ are all three members of the same binary recurrence sequence.

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

We prove that Hilbert’s Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$ ). We relate division-ample sets to arithmetic of abelian varieties.