We first discuss the Fermat-type equation with signature (2, m, n) which is the Diophantine equation in the shape x2 + ym = zn , where x, y and z are unknown integers, and m, n are fixed positive integers greater than 1. This kind of equations has been particularly focused on our work here in the case m = 2, n =5 and y = p is a fixed rational prime. Then the first result describing the condition of such a prime p in order to illustrate that this certain equation has an integer solution (x,y) when p ≡ 1(mod 4) and gcd (x, p) = 1, and the second result stating that the equation has no integer solution (x,y) when p ≡ 3(mod 4) are provided. Lastly, we will indicate that the results of Be′rczes and Pink about solving the equation x2 + p2k = zn in 2008 have been generalized in the particular cases (n,k) = (3,1) and (5,1), and additionally present that the first result and also its analogous result in the particular case n = 3 can be linked to the Bunyakovsky conjecture.
