ON THE DIOPHANTINE EQUATION x² + C = 2yⁿ

Abstract

Abstract. In this paper, we study the Diophantine equation x 2 + C = 2y n in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C ≡ 1 (mod 4) we explain how the equation can be solved using the multi-Frey variant of the modular approach. We illustrate our approach by solving completely the equations x 2 + 17 a 1 = 2y n , x 2 + 5 a 1 13 a 2 = 2y n , and x 2 + 3 a 1 11 a 2 = 2y n .

Complete solutions of certain Lebesgue--Ramanujan--Nagell type equations

“…A more general version of this case was studied by F. S. Abu Muriefah, F. Luca, S. Siksek and Sz. Tengely in [1] when D ≡ 1 (mod 4). On the other hand, for the case (m, λ) = (1, 4), F. Luca, Sz.…”

On the Diophantine equation 2m+nx2=yn

“…If is even, taking square-roots in Eq. (9) we get a = ±iα /2 , showing that i ∈ K. Hence, Q(i) ⊂ K. This is impossible for u > 1 (see Lemma 2.1 in [2] which asserts that Q( √ 2 ), Q(i √ u ) and Q(i √ 2u ) are the only quadratic fields in K).…”

On the solutions of a Lebesgue–Nagell type equation

“…A more general version of this case was closely studied by F. S. A. Muriefah, F. Luca, S. Siksek and Sz. Tengely in [20] when D ≡ 1 (mod 4). On the other hand for the case λ = 4, F. Luca, Sz.…”