A Diophantine Equation Integer Solution

The main work of this paper is:First,we get the condition that the equation 2py² = 2x³ + 3x² + x have no positive solution by the method of simple congruence and decomposition.Second, we discuss the problem of four class equation Dx² +1 = y^p (p≥5 and p is a prime ,D = 31,47, 71, 79) without nonzero integer solution. We obtain the following results:1. If p=5, we haveDx² + 1 = y⁵ (1)1). If 2| y, the equation (1) has non-zero integral solutions, then y > 20D + 1; 2). If 2 | y, let D = 31, 71, the equation (1) has non-zero integral solutions, then y≡2,8(mod31), y≡5,25(mod71) ; let D = 47,79, the equation (1) has no nonzero integral solutions.2). If p > 5, the equation Dx² + 1 = y^p has no non-zero integral solutions when D = 31,47, 71 (p≠7), 79 (p≠13).Third, consider the solvability of the equation x²- 2p = yⁿ. If the class number of Q[(2P)^2/1] is 1 or 2p - 1 = 3t, then1). If n = 3, then the equation has no solutions:2). There exists integerα₁≥2, the equation has no solutions for t≡1(mod 2^α1+1≡3(mod2^α1+1).