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Six Exponentials Theorem — Irrationality

Abstract : Let p, q, r be three multiplicatively independent positive rational numbers and u a positive real number such that the three numbers pu, qu, ru are rational. Then u is also rational. We prove this result by introducing a parameter L and a square L × L matrix, the entries of which are functions (ps1qs2rs3)(t0+t1u)x. The determinant Δ(x) of this matrix vanishes at a real point x ≠ 0 if and only if u is rational. From the hypotheses, it follows that Δ(1) is a rational number; one easily estimates a denominator of it. An upper bound for ∣Δ(1)∣ follows from the fact that the first L(L − 1)/2 Taylor coefficients of Δ(x) at the origin vanish.
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Submitted on : Tuesday, May 10, 2022 - 5:18:11 PM
Last modification on : Wednesday, May 11, 2022 - 3:48:15 AM

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Michel Waldschmidt. Six Exponentials Theorem — Irrationality. Resonance, Indian Academy of Sciences, 2022, 27, pp.599-607. ⟨10.1007/s12045-022-1351-0⟩. ⟨hal-03664120⟩

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