Solvability of the Diophantine equations xp + 22m y4 = pk z2 and xp + y2 = pk z4

We study the solvability of two classes of Diophantine equations by using some new methods and new results in this paper.Letp be an odd prime and Bn denote nth Bernoulli number.We prove that ifp ≡ 1(mod 4)andp |B(p-1)/2,then the equationxp +22mn4 = pky2,m,n,k ∈ N ,k ＞ 1,gcd(x,py)= 1,and the equationxp +y2 =pkz4,k ∈ N,gcd(x,y)= 1,k ＞ 1,21 y have no integral solutions respectively.