Application of difference sequences theory

TL;DR

This paper applies difference sequences theory to analytic functions and Diophantine equations, establishing a link between derivatives and difference sequences, and providing a method to identify regions where Diophantine equations lack integer solutions.

Contribution

It introduces a novel method connecting difference sequences with derivatives and Diophantine equations, improving the ability to find solution-free limits for higher power equations.

Findings

Derived an equation linking derivatives and difference sequences.

Identified limits where Diophantine equations have no integer solutions.

Method effectiveness increases with the power of the Diophantine equation.

Abstract

The results of difference sequences theory are applied to analytic function theory and Diophantine equations. As a result we have the equation which connects the $n$-th derivative of a function with the difference sequence for the values of this function. Also the results of difference sequences theory helps to discover some features of the whole kind of Diophantine equations. The method presented allows to find limits where Diophantine equation does not have integer solutions. The higher power of Diophantine equation the better this method works.