Perfect powers with few binary digits and related Diophantine problems

Abstract

We prove that, for any fixed base x ≥ 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.