Thue's Fundamentaltheorem, II: Further Refinements and Examples

TL;DR

This paper refines Thue's Fundamentaltheorem to derive effective irrationality measures for specific algebraic numbers generated by polynomial roots, particularly for cases where n=4 and 5, revealing infinitely many such numbers.

Contribution

It provides new, sharper bounds and simplified methods based on Thue's theorem for measuring irrationality of certain algebraic numbers.

Findings

Effective irrationality measures for roots of specific polynomial forms

Infinite algebraic numbers identified for n=4 and n=5

Simplified refinements of previous results

Abstract

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of polynomials of the form $(x-\sqrt{t})^{n}+(x+\sqrt{t})^{n}$, where $n \geq 4$ is a positive integer and $t$ is a negative integer. For $n=4$ and $n=5$, we find infinitely many such algebraic numbers.