(untitled) (Global Journal of Advanced Engineering Technologies and Sciences, 2023)
The present short paper, which is an amelioration of my previous article “confirmation of the Beal-BrunTijdeman-Zagier conjecture” published by the GJETS in 20/11/2019 [15], confirms the Beal’s conjecture, remained open since 1914 and saying that: "∃, , ∈ ℕ∗ such that ∶ + = and gcd, , = 1 ⇒ min , !, " = 2" The proof uses elementary tools of mathematics, such as the L’Hôpital rule, the Bolzano-Weierstrass theorem, the intermediate value theorem and the growth properties of certain elementary functions. The proof uses also the Catalan-Mihailescu theorem [18] [19] and some methods developed in my paper on the Fermat last theorem [14] published by the GJAETS in 10/12/2018. The particular case of the Fermat last theorem is deduced.