Rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ f
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Rational solutions of the Diophantine equations f (x)2 ± f (y)2 = z 2 Ahmed El Amine Youmbai1,2
· Djilali Behloul1,3
© Akadémiai Kiadó, Budapest, Hungary 2019
Abstract We discuss the rational solutions of the Diophantine equations f (x)2 ± f (y)2 = z 2 . This problem can be solved either by the theory of elliptic curves or by elementary number theory. Inspired by the work of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201, 2010) and following the approach of Zhang and Zargar (Period Math Hung, 2018. https://doi.org/10. 1007/s10998-018-0259-7) we improve the results concerning the rational solutions of these equations. Keywords Diophantine equations · Rational solutions · Elliptic curves · Conic sections Mathematics Subject Classification Primary 11D45 · 11D41 · Secondary 14H52
1 Introduction and results Let f (x) ∈ Q[x] be a polynomial without multiple roots and consider the Diophantine equations z 2 = f (x)2 + f (y)2 (1.1) and z 2 = f (x)2 − f (y)2 .
(1.2)
The problem dates back to 2010 when Ulas and Togbé in [4] showed that if f (x) has degree 2 then the set of rational parametric solutions of the equations z 2 = f (x)2 ± f (y)2
B
(1.3)
Ahmed El Amine Youmbai [email protected] Djilali Behloul [email protected]
1
LATN Laboratory, Faculty of Mathematics, USTHB, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria
2
Mathematics Department, Faculty of Exact Sciences, University of El Oued, PO Box 789, 39000, El Oued, Algeria
3
Computer Science Department, USTHB, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria
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A. E. A. Youmbai, D. Behloul
is non-empty and if deg( f ) = 3 and f has the form f (x) = x(x 2 + ax + b) with a = 0, then (1.3) has infinitely many non-trivial rational parametric solutions. They also proved the same result for more general cubic polynomials of the form f (x) = x 3 + ax 2 + b with b = 0 for Eq. (1.2) and asked whether there exists a polynomial of higher degree such that one or both Eqs. (1.1) and (1.2) are satisfied for infinitely many non-trivial rational solutions (x, y, z). We also mention a closely related paper by Tengely and Ulas [3] in which they studied the existence of integral solutions of the Diophantine equations z 2 = f (x)2 ± g(y)2 for certain polynomials f , g ∈ Z[x] of degree at least 3. Recently in October 2018, Zhang and Zargar [6] answered the question of Ulas and Togbé using quartic polynomials. In this paper, we improve some results in [6] by proving for some polynomials of degree 2n + 3 (n ∈ N) that both Eqs. (1.1) and (1.2) have infinitely many non-trivial rational solutions parameterized by certain quartic elliptic curves of positive rank. Then, using an elementary method, we generalize this result and solve the problem in a different way using polynomials f (x) of any degree n for both Eqs. (1.1) and (1.2) by showing that there exist infinitely many non-trivial rational solutions parametrized by some conic sections. The result, though readily obtained, is believed to be new. In other words we prove the following: Theorem 1.1 Le
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