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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\left( 4^{q}-1\right) ^{u} +\left( 2^{q+1}\right) ^{v}=w^{2}$},
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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/

[1] L.Jésmanowicz, “Several remarks on Pythagorean numbers”, Wiadom. Mat., 1 (1955/56), 196–202 | MR

[2] W.Sierpinski, “On the equation $3^{x}+4^{y}=5^{z}$”, Wiadom. Mat., 1 (1955/56), 194–195 | MR

[3] W.Sierpinski, Elementary Theory of Numbers, PWN-Polish Scientific Publishers, Warszawa, 1988 | MR | Zbl

[4] N.Terai, “The Diophantine equation $x^{2}+q^{m}=p^{n}$”, Acta Arith., 63:4 (1993), 351–358 | DOI | MR | Zbl

[5] M.Le, G.Soydan, “A note on Terai's conjecture concerning primitive Pythagorean triples”, Hacettepe Journal of Mathematics and Statistics, 50:4 (2021), 911–917 | DOI | MR | Zbl

[6] Z.Xinwen, Z.Wenpeng, “The Exponential Diophantine Equation $ \left( (2^{2m}-1)n\right) ^{x}+\left( 2^{m+1}n\right) ^{y}=((2^{2m}+1)n)^{z}$”, Bulletin Mathématique de La Société Des Sciences Mathé matiques de Roumanie, 57:3(105) (2014), 337–44 | MR

[7] H.Yang, R.Fu, “A Further Note on Jésmanowicz' Conjecture Concerning Primitive Pythagorean Triples”, Mediterr. J. Math., 19 (2022), 57 | DOI | MR

[8] T.Miyazaki, “Generalizations of classical results on J ésmanowicz' conjecture concerning Pythagorean triples”, AIP Conference Proceedings, 1264, 2010, 41–51 | DOI | MR | Zbl

[9] T.Miyazaki, “Terai's conjecture on exponential Diophantine equations”, Int. J. Number Theory, 7:4 (2011), 981–999 | DOI | MR | Zbl

[10] T.Miyazaki, “Exceptional cases of Terai's conjecture on Diophantine equations”, AIP Conference Proceedings, 1385, 2011, 87–96 | DOI | MR | Zbl

[11] T.Miyazaki, “Exceptional cases of Terai's conjecture on Diophantine equations”, Arch. Math., 95 (2010), 519–527 | DOI | MR | Zbl

[12] N.Terai, Yo.Shinsho, “On the exponential Diophantine equation $(3m^2+1)^x+(qm^2-1)^y=(rm)^z $”, SUT J. Math., 56:2 (2020), 147–158 | DOI | MR | Zbl

[13] N.Terai, Yo.Shinsho, “On the exponential Diophantine equation $(4m^2+1)^x+(45m^2-1)^y=(7m)^z $”, International Journal of Algebra, 15:4 (2021), 233–241 | DOI | MR