Quantitative indicators of the solutions of Diophantine equations and systems in the domain of the natural numbers

TL;DR

This paper investigates the asymptotic density of solutions to Diophantine equations in natural numbers, providing estimation methods and bounds for the number and probability of solutions across various types and orders of equations.

Contribution

It introduces new estimation techniques for the number, density, and probability of solutions to Diophantine equations and systems, including geometric proofs for second-order cases.

Findings

Asymptotic density of solutions is zero.

Provides bounds for the number of solutions of various Diophantine equations.

Establishes geometric estimates for second-order equations.

Abstract

The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $<x_1,...x_k>$ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometrically proved in the paper. The author proves the assertion about the number of solutions of algebraic Diophantine equations of higher orders in the domain of the natural numbers. The author provides the estimates for the asymptotic behavior of quantitative solutions of Diophantine equations and systems in the domain of the natural numbers.