Perfect powers with few binary digits and related Diophantine problems

Bennett, Michael A.; Bugeaud, Yann; Mignotte, Maurice

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 941-953.

We prove that, for any fixed base $x\ge 2$ and sufficiently large prime $q$, no perfect $q$-th power can be written with $3$ or $4$ digits $1$ in base $x$. This is a particular instance of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms.

Classification : 11A63, 11D61, 11J86

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