Non-archimedean analysis on the extended hyperreal line $^*R_d$ and the solution of some very old transcendence conjectures over the field $Q$

TL;DR

This paper extends non-archimedean analysis to the hyperreal line $^*R_d$, proving new transcendence results for numbers like $e+\pi$, $e\pi$, and $e^e$, and generalizing classical theorems in number theory.

Contribution

It introduces a completion of the non-archimedean field and applies it to solve longstanding transcendence conjectures over the rationals.

Findings

Proves $e+\pi$ and $e\pi$ are irrational.

Shows $e^e$ is transcendental.

Generalizes Lindemann-Weierstrass theorem.

Abstract

In this paper possible completion $^*R_{d}$ of the Robinson non-archimedean field $^*R$ constructed by Dedekind sections. Given an class of analytic functions of one complex variable $f \in C[z]$,we investigate the arithmetic nature of the values of $f$ at transcendental points $e^{n}$. Main results are: 1) the both numbers $e+\pi$ and $e\pi$ are irrational, 2) number $e^{e}$ is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtainedNon conservative extensions of the canonical nonstandard analysis also is considered.