This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with ...

We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it “pellans sequence” and then we solve the system ab+ 1 = Sx, ac + 1 = Sy bc+ 1 = Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

With my appreciation, I wish to thank my advisor Prof. Hillel Furstenberg for his guidance, support and encouragement during these unforgettable years of learning and research Abstract We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain " algebraic patterns " in every s...

DIOPHANTINE EQUATIONS INVOLVING ARITHMETIC FUNCTIONS OF FACTORIALS Daniel M. Baczkowski We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation:

The Subspace Theorem, whose name will be clear from its statement, was proved by Wolfgang Schmidt around forty years ago. It provides us with a multidimensional extension of Roth’s Theorem and was originally developed for the study of two classical problems, namely algebraic approximation to algebraic numbers and norm form equations (a class of Diophantine equations which includes the Thue equa...

In recent years, we have seen several approaches to the graph isomorphism problem based on “generic” mathematical programming or algebraic (Gröbner basis) techniques. For most of these, lower bounds have been established. In fact, it has been shown that the pairs of nonisomorphic CFI-graphs (introduced by Cai, Fürer, and Immerman in 1992 as hard examples for the combinatorial Weisfeiler-Leman a...

Natural phenomenon of coevolution is the reciprocally induced evolutionary change between two or more species or population. Though this biological occurrence is a natural fact, there are only few attempts to use this as a simile in computation. This chapter is an attempt to introduce reciprocally induced coevolution as a mechanism to counter problems faced by a typical genetic algorithm applie...

We describe through an algebraic and geometrical study, a new method for solving systems of linear diophantine equations. This approach yields an algorithm which is intrinsically parallel. In addition to the algorithm, we give a geometrical interpretation of the satissability of an homogeneous system, as well as upper bounds on height and length of all minimal solutions of such a system. We als...

Assuming the existence of a solution, we find bounds for small solutions x of the finite matrix equation Ax = B, where each entry of A, B is an integer, and x is a nontrivial column vector with nonnegative integer entries. 0. Introduction. In [1], [5] and [6] there arise in a topological setting, systems of linear equations with integer coefficients. The problem is to find a bound K depending o...