Homogeneous and nonhomogeneous Diophantine equations

Linear Homogeneous Diophantine Equations and Magic Labelings of Graphs

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...

Diophantine Approximations, Diophantine Equations, Transcendence and Applications

This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...

Diophantine Equations and Congruences

We present conditions for quadratic Diophantine equations of the form ax2 − by2 = ±1, (where 1 < a < b are integers) for which there are no solutions (x, y), yet for which there are solutions modulo n for all n ≥ 1. This generalizes work in the literature which follow as very special cases. Mathematics Subject Classification: Primary: 11D09, 11R11, 11A55; Secondary: 11R29

{nkx} and diophantine equations

We establish a law of the iterated logarithm for the discrepancy of sequences (nkx) mod 1 where (nk) is a sequence of integers satisfying a sub-Hadamard growth condition and such that one and four-term Diophantine equations in the variables nk do not have too many solutions. The conditions are discussed, the probabilistic details of the proof are given elsewhere. As a corollary to our results, ...