Geodesic
A note on the Diophantine equation $\left( 4^{q}-1\right) ^{u} +\left( 2^{q+1}\right) ^{v}=w^{2}$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 2, pp. 275-278

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Let $a, b$ and $ c $ be positive integers such that $a^{2}+b^{2}=c^{2}$ with $\gcd \left( a,b,c\right) =1$, $a$ even. Terai's conjecture claims that the Diophantine equation $x^{2}+b^{y}=c^{z}$ has only the positive integer solution $(x,y,z)=(a,2,2)$. In this short note, we prove that the equation of the title, has only the positive integer solution $(u,v,w)=(2,2,4^{q}+1),$ where $q$ is a positive integer.
Keywords: Pythagorean triple.
Mots-clés : Terai's conjecture 
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     title = {A note on the {Diophantine} equation $\left( 4^{q}-1\right) ^{u} +\left( 2^{q+1}\right) ^{v}=w^{2}$},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
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Djamel Himane; Rachid Boumahdi. A note on the Diophantine equation $\left( 4^{q}-1\right) ^{u} +\left( 2^{q+1}\right) ^{v}=w^{2}$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 2, pp. 275-278. http://geodesic.mathdoc.fr/item/JSFU_2023_16_2_a12/