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 \({\left( {{p^{{s_1}}}{q^{{s_2}}}{r^{{s_3}}}} \right)^{\left( {{t_0} + {t_1}u} \right)x}}\). 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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K. Senthil Kumar, Veekesh Kumar and R. Thangadurai, On a Problem of Alaoglu and Erdős, Resonance, Vol.23, No.7, pp.749–758, 2018. https://www.ias.ac.in/article/fulltext/reso/023/07/0749-0758
S. Lang, Introduction to Transcendental Numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.
M. Laurent, Sur quelques résultats récents de transcendance, Astérisque, No.198–200, pp.209–230, 1992; Journées Arithmétiques, Luminy 1989. https://smf.emath.fr/publications/sur-quelques-resultats-recents⌕otect@normalcrrelax-de-transcendance.
K Ramachandra, Contributions to the theory of transcendental numbers— I, II, Acta Arith., 14, pp.65–72 and pp.73–88, 1967/68. https://doi.org/10.4064/aa-14-1-73-88
M. Waldschmidt, Linear independence of logarithms of algebraic numbers, IMS Report, Vol.116, Institute of Mathematical Sciences, Madras, 1992, With an appendix by Michel Laurent. https://webusers.imj-prg.fr/∼michel.waldschmidt/articles/pdf/LIL.pdf
M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups, Grundlehren der Mathematischen Wissenschaften, Vol.326, Springer-Verlag, Berlin, 2000. DOI: https://doi.org/10.1007/978-3-662-11569-5
H. Halberstam, Transcendental numbers, Math. Gaz., 58, pp.276–284, 1974. DOI: https://www.jstor.org/stable/3616099
L. Alaoglu, and F. Erdös, On highly composite and similar numbers, Trans. Amer. Math. Soc., Vol.56, pp.448–469, 1944. DOI: https://doi.org/10.2307/1990319
D. C. Cantor and E. G. Straus, On a conjecture of D. H. Lehmer, Acta Arith., Vol.42, No.1, pp.97–100, Correction Id. No.3, p.327, 1982/83. DOI: https://doi.org/10.4064/aa-42-1-97-100.
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Michel Waldschmidt is a French mathematician specializing in number theory, in particular, Diophantine problems, which includes transcendental number theory. He is presently an Emeritus Professor at Sorbonne University.
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This text is mainly an English translation by the author of his paper published in French in the Revue de la Filière Mathématiques-RMS with the title ‘Le théorème des six exponentielles restreint à l’irrationalité’.
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Waldschmidt, M. Six Exponentials Theorem — Irrationality. Reson 27, 599–607 (2022). https://doi.org/10.1007/s12045-022-1351-0
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DOI: https://doi.org/10.1007/s12045-022-1351-0