Finiteness of odd perfect powers with four nonzero binary digits

We prove that there are only finitely many odd perfect powers in N having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at S-unit points (in a suitable -adic convergence), Roth's general theorem, 2-adic Padé approximations (by integers) to numbers in varying number fields and lower bounds for linear forms in two logarithms (both in the usual and in the 2-adic context). © Annales de L'Institut Fourier.