Adamczewski, Boris; Bugeaud, Yann Real and \(p\)-adic expansions involving symmetric patterns. (English) Zbl 1113.11041 Int. Math. Res. Not. 2006, No. 14, Article ID 75968, 17 p. (2006). Inspired by a conjecture of Mahler and of Mendès France, the authors state the following problem: let \((a_n)_{n\geq 1}\) be a sequence of integers with values in \([0,p-1]\). Is it true that the real number \(\sum a_n p^{-n}\) and the \(p\)-adic number \(\sum a_n p^n\) are both algebraic if and only if they are both rational? While this question remains open, they prove that the answer is positive for the sequence \((a_n)_{n\geq 1}\) containing infinitely many palindromes “not too far’ from its beginning (their statement is of course quantitative, and more general). Please note that reference [2] appeared [B. Adamczewski and Y. Bugeaud, Ann. Math. (2) 165, 547–565 (2007; Zbl 1195.11094)]. Reviewer: Jean-Paul Allouche (Orsay) Cited in 6 Documents MSC: 11J81 Transcendence (general theory) 68R15 Combinatorics on words Keywords:Mahler conjecture; transcendence of real and \(p\)-adic numbers; palindromes; Schmidt-Schlickewei subspace theorem Citations:Zbl 1195.11094 PDFBibTeX XMLCite \textit{B. Adamczewski} and \textit{Y. Bugeaud}, Int. Math. Res. Not. 2006, No. 14, Article ID 75968, 17 p. (2006; Zbl 1113.11041) Full Text: DOI