Diophantus of Alexandria was a greek mathematician, around 200 AD, who studied mathematical problems, mostly geometrical ones, which he reduced to equations in rational integers or rational numbers. He was interested in producing at least one solution. Such equations are now called Diophantine equations. An example is y − x = 1, a solution of which is (x = 2, y = 3). More generally, a Diophanti...

A -system consists of quadratic Diophantine equations over nonnegative integer variables of the form:

In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteri...

We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Speci cally, we prove that there are only ve types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizat...

Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s ...

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These are inspired by Rado's characterization regular linear homogeneous equations. We conjecture that these also sufficient regularity, at least equations whose corresponding monovariate polynomial is linear. This would natural generalization the...

We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is asymptotically solution to a linear Boltzmann equ...

The rational solutions with as denominators powers of to the elliptic diophantine equation y x x are determined An idea of Yuri Bilu is applied which avoids Thue and Thue Mahler equations and deduces four term S unit equations with special properties that are solved by linear forms in real and p adic logarithms Introduction In a recent paper SW my colleague R J Stroeker and I determined the com...