Linear Independence Measures for Logarithms of Algebraic Numbers

Waldschmidt, Michel

doi:10.1007/3-540-44979-5_5

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Proof Of Proposition 131

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“…Specifically, since clearly log |y|, log 2 are linearly independent over Q, we can apply Theorem 5.1, p. 317 of [8]. (10) We choose the following data to be inserted in that general statement:…”

Section: Proof Of Proposition 13mentioning

confidence: 99%

Finiteness of odd perfect powers with four nonzero binary digits

Corvaja¹,

Zannier²

2013

Ann. inst. Fourier

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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

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Section: Proof Of Proposition 13mentioning

confidence: 99%