On the solutions of a Lebesgue–Nagell type equation

Abstract

We find all positive integer solutions in x, y and n of x 2 + 19 2k+1 = 4y n for any non-negative integer k.

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

“…in non-negative integers x, y, m and n for p ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163}. We note that (1.5) has been solved completely for p = 19 in [8]. It is easy to see that (1.5) has no solution for p = 1.…”

“…The remaining values of C in this range were covered in [8,25] (when λ = 1). There are numerous results for λ = 2, 4, and interested readers can look into [6,22,26] and references therein. In the recent years, several authors became interested in the case when C = p a 1 1 p a 2 2 • • • p a k k , where p i 's are distinct primes and k ≥ 1, a i ≥ 0 are integers.…”

On the Diophantine Equation $$cx^2+p^{2m}=4y^n$$

“…where c and d are given positive integers, have been considered by several authors over the decades. In particular, there are many interesting results about the integer solutions of this equation for d = 1 and we direct the reader to the papers [2,7,13,16,17] for more information. For a survey on this very interesting subject we recommend [15,19].…”

The Diophantine equation x² + 3ᵃ · 5ᵇ · 11ᶜ · 19ᵈ = 4yⁿ

“…It has been studied by several authors. Luca, Tengely, and Togbé [7] studied (1.1) when 1 ≤ ≤ 100 and ̸ ≡ 1 (mod 4), = 7 • 11 , or = 7 • 13 , where , ∈ N. Bhatter, Hoque, and Sharma [1] studied (1.1) when = 19 2 +1 , where ∈ N. Chakraborty, Hoque, and Sharma [4] studied (1.1) when =…”

“…, where 2 ∈ N. Then (3.41) reduces to )︀ = −1. Case 6.3: ( 1 , 2 , 3 , 4 ) = (1,5,5,3). Then (3.33) reduces to…”