This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...

In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations, relating these solutions by means of particular linear recurrence sequences. We point out interesting relationships between these sequences and known sequences i...

The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert‟s tenth problem. The proposed methodology uses a three layer feed forward neural network with back propagation as sequential learning procedure to find numerical solutions of a class of Diophantine equations. It uses a dynamically constructed net...

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

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions: (i) Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem? (ii) Is it useful to combine this approach with traditional approaches to Diophantine e...

The complete sets of solutions of the equation (n k ) = (m ` ) are determined for the cases (k, `) = (2, 3), (2, 4), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k, `) = (3, 6)) we h...

The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For n = 2 such a procedure is well known, when new variables are components of spinors and they are widely used in mathematical physics. For example, parametrization of Pythagoras threes a + b , a − b , 2ab may be cited as an example in number theory where two independ...

The three numbers 1, 5, and 442 have the property that the product of any two numbers decreased by 1 is a perfect square. In this paper it is proved that there is no other positive integer which shares this property with 1, 5, and 442. Mathematics Subject Classification: Primary 11D09, 11D25, Secondary 11B37, 11J68, 11J86, 11Y50.

We relate a previous result of ours on families of diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation, on the one hand with a Liouville type estimate, on the other hand with an estimate arising from a lower bound for a linear combination of logarithms. American Mat...

New algorithms are described and analysed for solving various problems associated with a large integer matrix: computing the Hermite form, computing a kernel basis, and solving a system of linear diophantine equations. The algorithms are space-efficient and for certain types of input matrices — for example, those arising during the computation of class groups and regulators — are faster than pr...