The many faces of the Kempner number

TL;DR

This survey reviews five different proofs of the transcendence of Kempner's number, highlighting various methods and ideas used in transcendence proofs and related results.

Contribution

It provides a comprehensive overview of multiple proofs for Kempner's number transcendence, emphasizing diverse techniques and recent developments.

Findings

Five distinct proofs of transcendence presented

Different mathematical techniques discussed

References for detailed proofs provided

Abstract

In this survey, we present five different proofs for the transcendence of Kempner's number, defined by the infinite series $\sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas and methods that are used for proving deeper results. We outline proofs for some of these results and also point out references where the reader can find all the details.