Complexities Approach to Two Problems In Number Theory

TL;DR

This paper applies Kolmogorov Complexity to address two longstanding number theory problems, offering a novel approach that bridges computational complexity and traditional mathematics.

Contribution

It introduces a new method using Kolmogorov Complexity to solve problems in number theory, expanding the application of computational concepts beyond logical disciplines.

Findings

Solved the problem of identifying non-period numbers using Kolmogorov Complexity.

Addressed the existence of bounded continued fraction coefficients for transcendental numbers.

Proposed a new perspective linking computational complexity with classical mathematical problems.

Abstract

By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 \emph{Exhibit at least one number which does not belong to} $ \mathcal{P}$ (period number) in their paper,another is the problem about existence of bounded coefficients of continued fraction expansion of transcendental number.Thus we show a new approach to mathematical problems in the non-logical discipline.Futhermore,we show that resource-bounded Kolmogorov Complexity and computational complexity can at least provide tips or principles to mathematical problems in the non-traditional or logical discipline.