Let p is a prime, we studied the the generalized Lebesgue-Ramanujan-Nagell equation. By using the elementary method and algebraic number theory, we obtain one necessary condition which the equation has integer solutions and some sufficient conditions which the equation has no integer solution. 1). Let x be an odd number, one necessary condition which the equation has integer solutions is that 2 n(p-1) -1/p contains some square factors. 2). Let x be an even number, when n=pk(k≥1), all integer

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... utions for the equation are (x,y)=(0,4 k ); when n=pk+(p-1)/2(k ≥0), all integer solutions are (±2 pk+(p-1)/2 ,2 2k+1 ); when n≡1,2,3,...,(p-3)/2, (p+1)/2,...,p-1(modp), the equation has no integer solution.