Finiteness of odd perfect powers with four nonzero binary digits

Corvaja, Pietro; Zannier, Umberto

Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 715-731.

Nous démontrons la finitude de l’ensemble des puissances pures impaires ayant quatre chiffres non nuls dans leur écriture binaire. La preuve de ce théorème amène naturellement à des énoncés plus généraux, mais, pour simplifier, nous avons préféré nous borner à ce résultat. Notre méthode combine plusieurs ingrédients  : des résultats (dérivés du théorème du sous-espace) sur les valeurs entières de séries analytiques aux points $S$-unités, le théorème de Roth généralisé, les approximations de Padé $2$-adiques de nombres algébriques dans un corps variable, des minorations de formes linéaires en deux logarithmes (par rapport aux valeurs absolues archimédiennes et $2$-adique).

We prove that there are only finitely many odd perfect powers in $\mathbb{N}$ 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 $\nu$-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).

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