On The Diophantine Equation X~2+5=P~n

The Diophantine Equations not only developed actively itself, but also were applied to other fields of Discrete Mathematics. They play an important role in people' study and research .So many researchers study the Diophantine Equation extensively and highly in the domestic and abroad. Along with the development of the Diophantine Equation, Algebraic Number obtained the primary formation and development. Especial after the mathematician Kummer introduced the concept of ideal, the study of Diophantine Equations has a great breakthrough.As an important part of Algebraic Number, quadratic fields ,the arithmetic in quadratic fields ,ideal ,class number and unique factorization theory play an important role in research of Diophantine Equation. For some D, as for the integer solution of Diophantine Equation x~2+ 5 = p~n, if the class number of Q((?)) is 1, we will directly apply its corresponding unique factorization theory in algebraic integer domain, if the class number of quadratic fields Q((?)) is not 1, we will first apply its corresponding unique factorization theory of ideal in algebraic integer domain ,then apply the property of principal ideal domain being unique factorization domain to solve.In this paper , we will prove all the integer solutions of the Diophantine Equation. x~2+5 = p~n(0