CIRM - Videos & books Library - Number of solutions to a special type of unit equations in two unknowns

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Résumé : For any fixed coprime positive integers a, b and c with min{a, b, c} > 1, we prove that the equation $a^{x}+b^{y}=c^{z}$ has at most two solutions in positive integers x, y and z, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equations of S.S.Pillai, Canad. J. Math. 53(2001), no.2, 897–922] which asserts that Pillai's type equation $a^{x}-b^{y}=c$ has at most two solutions in positive integers x and y for any fixed positive integers a, b and c with min {a, b} > 1. In this talk we give a brief summary of corresponding earlier results and present the main improvements leading to this definitive result. This is a joint work with T. Miyazaki.

Keywords : Pillai's equation; S-unit equation; Barber's method

Codes MSC :
11D41 - "Higher degree equations; Fermat's equation"
11D45 - Counting solutions of diophantine equations
11D61 - Exponential equations

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https://videos.cirm-math.fr/2020-11-26_Pink.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Dates : 23/11/2020 - 27/11/2020
Année de la rencontre : 2020
URL Congrès : https://www.chairejeanmorlet.com/2256.html

Données de citation

DOI : 10.24350/CIRM.V.19688803
Citer cette vidéo: (2020). Number of solutions to a special type of unit equations in two unknowns. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19688803
URI : http://dx.doi.org/10.24350/CIRM.V.19688803

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Bibliographie

BENNETT, Michael A. On some exponential equations of SS Pillai. Canadian Journal of Mathematics, 2001, vol. 53, no 5, p. 897-922. - https://doi.org/10.4153/CJM-2001-036-6

Hu, Yongzhong; Le, Maohua; An upper bound for the number of solutions of ternary purely exponential Diophantine equations II. Publ. Math. Debrecen 95 (2019), no. 3-4, 335–354. - https://doi.org/10.5486/PMD.2019.8444

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