DIOPHANTINE EQUATIONS IN CERTAIN RINGS

On Certain Diophantine Equations in Rings and Fields.

A New Method for Certain Diophantine Equations

1. Introduction. The original purpose of the research described in this article was to obtain results about Diophantine problems on rational surfaces — that is, surfaces defined over a field k which are birationally equivalent to P 2 over the algebraic closure ¯ k. (Throughout this article, K and k will always denote algebraic number fields, with respective rings of integers O and o. Except in ...

On Certain Diophantine Equations Related to Triangular and Tetrahedral Numbers

In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians NUMBER THEORY, Kraków, 14th20th september 2008.

Nonnegative Solvability of Linear Equations in Certain Ordered Rings

In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered ...

Diophantine approximation and Diophantine equations

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...