Adamczewski, Boris The many faces of the Kempner number. (English) Zbl 1286.11109 J. Integer Seq. 16, No. 2, Article 13.2.15, 34 p. (2013). In this survey the author shows several methods to prove that the Kempner’s number \(\sum_{n=1}^\infty 2^{-2^n}\) is transcendental. Each method is accompanied with some interesting comments. Reviewer: Jaroslav Hančl (Ostrava) Cited in 3 Documents MSC: 11J81 Transcendence (general theory) 11J87 Schmidt Subspace Theorem and applications 11J70 Continued fractions and generalizations Keywords:Transcendence; Kempner number; diophantine approximation; radix expansion; continued fraction; Mahler’s method; subspace theorem Software:OEIS PDFBibTeX XMLCite \textit{B. Adamczewski}, J. Integer Seq. 16, No. 2, Article 13.2.15, 34 p. (2013; Zbl 1286.11109) Full Text: arXiv EMIS Online Encyclopedia of Integer Sequences: Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n). Decimal expansion of Sum_{n>=0} 1/2^(2^n). Decimal expansion of C = Sum_{k>=0} 1/2^(2^k-1). Decimal expansion of Sum_{k>=0}((-1)^k/2^(2^k)).