On the Diophantine equation $$Cx^{2}+D=2y^{q}$$ C x 2 + D = 2 y q
- PDF / 277,170 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 99 Downloads / 201 Views
On the Diophantine equation Cx 2 + D = 2y q Nejib Ghanmi1
· Fadwa S. Abu Muriefah2
Received: 29 October 2018 / Accepted: 15 March 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let C and D denote positive integers such that C D > 1. In this paper we investigate the solvability of the Diophantine equation C x 2 + D = 2y q , in positive integers x, y and odd prime number q where C D ≡ 3 (mod 4) and C D is squarefree. Keywords Diophantine equation · Fibonacci sequence · Primitive divisor · Lehmer pair · Lucas sequence Mathematics Subject Classification Primary 11D41 · Secondary 11D61, 11Y50
1 Introduction Let√C, D, n, h be positive integers where h denotes the class number of the field Q( −C D). The Diophantine equation C x 2 + D = 2y n ,
(1)
where gcd(h, n) = 1, has been studied by several authors over the years. The first general result is due to Ljunggren [7], who described the solution of Eq. (1) when certain conditions are satisfied. In 1976, Schinzel and Tijdeman [11] proved that for given non-zero integers C and D, Eq. (1) has only a finite number of solutions (x, y, n) which can be effectively determined. When C = D = 1, Eq. (1) was solved by Cohn in [4]. Pink and Tengely resolved the equation x 2 + a 2 = 2y n , with 1 ≤ a ≤ 1000 and 3 ≤ n ≤ 80 in their paper [9]. In [12] solutions were found by Tengely for C = 1 and for all values D which are the square of an integer B ∈ {3, 5, 7, . . . , 501}. The same
B
Fadwa S. Abu Muriefah [email protected]; [email protected] Nejib Ghanmi [email protected]; [email protected]
1
Department of Mathematics, University College of Jammum, Mecca, Saudi Arabia
2
Department of Mathematics, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia
123
N. Ghanmi, F. S. Abu Muriefah
author gave in [13] a new result for Eq. (1) when C = 1, D = p 2m , m > 0, and p an odd prime with gcd(x, y) = 1. Moreover, he proved that there are only finitely many solutions for which y is not a sum of consecutive squares. In 2009, Abu Muriefah et al. solved the equation x 2 + D = 2y n for a fixed D and n ≥ 3 under some restrictions (see [1] for details). In [14], the authors extended the work of Tengely and described the solutions of the equation x 2 + p 2 = 2y n . In fact, literature is very rich and there are many papers studying the Diophantine equation (1) for many special cases. In this paper we investigate the solvability of the Diophantine equation C x 2 + D = 2y q ,
(2)
where q is an odd prime, C D is squarefree, C D > 1, C D ≡ 3 (mod 4), and gcd(h, q) = 1. Our work is a generalization of that of Ljunggren [7] where we apply the powerful result of Bilu, Hanrot, and Voutier on primitive divisor of Lehmer sequence (see [2] for details). We prove the result given by Theorem 1 and we display in a separate section the solutions of some Diophantine equations. Theorem 1 Suppose that q is an odd prime, C D is squarefree, C D > 1, C D ≡ 3 (mod 4), and gcd(h, q) = 1. Then the Diophantine equation (2) has solutions only in th
Data Loading...
