On certain Diophantine equations of the form z2=f(x)2±g(y)2

Tengely, Sz.; Ulas, Maciej

doi:10.1016/j.jnt.2016.10.014

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Introduction1

For F (X) =1

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Cited by 10 publications

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“…(1.1) and (1.2), in 2017, Sz. Tengely and M. Ulas [8] investigated the existence of the non-trivial integer solutions of the Diophantine equations z 2 = f (x) 2 ±g(y) 2 and proved similar results for some special higher degree polynomials as well.…”

Section: Introductionmentioning

confidence: 70%

On the Diophantine equations z^2=f(x)^2 ± f(y)^2 involving Laurent polynomials II.

Zhang¹,

Tang²,

Zhang³

2022

MMN

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By the theory of Pell's equation, we give conditions for f (x) = b + c x with b, c ∈ Z\{0} such that the Diophantine equations z 2 = f (x) 2 ± f (y) 2 have infinitely many solutions x, y ∈ Z and z ∈ Q, which gives a positive answer to Question 3.2 of Zhang and Shamsi Zargar [16]. By the theory of elliptic curve, we study the non-trivial rational solutions of the above Diophantine equations for Laurent polynomials f, and give a positive answer to Question 3.1 of Zhang and Shamsi Zargar [16].

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Section: Introductionmentioning

confidence: 70%