On the diophantine equation x 2 + 2a · 3b · 11c = y n

Abstract

In this note, we find all the solutions of the Diophantine equation x 2 + 2a · 3b · 11c = y n, in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.

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

“…A natural next step in this series of explorations is to consider (1.1) when C has three distinct prime factors. Recent progress in this case were made in [10,14,15,16] for C = 2 a 3 b 11 c , 2 a 3 b 13 c , 2 a 3 b 17 c , 2 a 13 b 17 c , 2 a 5 b 13 c .…”

On the diophantine equation $x^2+2^a3^b73^c=y^n $

“…With the development of modern tools such as primitive divisor theorem, modular approach, or computational techniques, many authors investigated the above equation when k ≥ 1, see for example [1,2,3,7,11,13,14,15,16,17,18,19,20,22]. Especially, the cases (p 1 , p 2 , p 3 ) = (2,3,11), (2,11,19), (2,3,19), (2,3,17), (5,13,17) are considered in [6,9,10,11] and [21], respectively. For more information and the rich literature on equation (1), we refer to an excellent survey [12] and the 359 references therein.…”

On the Diophantine equation $x^2 + 7^\alpha \cdot 11^\beta = y^n$

“…In 2008, the equations x 2 C 5 a :13 b D y n and x 2 C 2 a 5 b :13 c D y n were solved in [5] and [21]. Recently, in [14] and [13], complete solutions of the equations x 2 C 2 a :11 b D y n and x 2 C 2 a :3 b :11 c D y n were found. In [20], the complete solution .n; a; b; x; y/ of the equation x 2 C 5 a :11 b D y n when gcd.x; y/ D 1; except for the case when xab is odd, is given.…”

“….i / If˛is even andˇD 0; then the only integer solutions of the Diophantine equation Proof. See [29], [12] and [14].…”