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

Abstract

It is well-known that for p = 1, 2, 3, 7, 11, 19, 43, 67, 163, the class number of Q(√ −p) is one. We use this fact to determine all the solutions of x 2 + p m = 4y n in non-negative integers x, y, m and n.

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].…”

“…Let d ∈ {2, 3, 7, 11, 19, 43, 67, 163} and p > 41 be a prime. Then (d + p) p ≡ d (mod p) and we have: 3,7,11,19,43, 67, 163} and p > 41 be a prime such that d + p is also a prime. Then the Diophantine equation 11,15,19,35,39,43,51,55,67,91,95,111,115,123,155,163,183,187,195,203…”

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

“…The proof of Theorem 1.1 is now achieved by means of the following four lemmas. We only require the condition ( , , , ( , , , , , , ) ( , , , , , , 4 . From (3.1) we have .…”

“…Case 3: = 55. Then ( 1 , 2 , 3 , 4 ) = (0, 1, 5, 4), (4, 1, 3, 4), (4, 5, 3, 4), (4, 5, 5, 2), (4,5,5,4). Equation (3.6) reduces to )︀ = −1.…”

“…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 =…”