Abstract—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 solutions for the equation are ( x , y)=(0,4^k); when   n  = pk +(p-1) /2( k≥0), all integer solu tions are (±2^pk+(p-1)/2,2^2k+1); when n≡1,2,3,…,(p-3)/2, ( p +1)/2,…,p-1(modp), the equati on has no integer solution.

Index Terms—Exponential Diophantine equation, integer solutions, integer ring, algebraic number theory.

Ran Yinxia is with Longnan Teachers College, Department of Mathematics, Gansu Longnan, China (email: ranyinxia@163.com).

Cite: Ran Yinxia, "On the Generalized LEBESGUE-Ramanujan-Nagell Equation," International Journal of Applied Physics and Mathematics vol. 7, no. 2, pp. 112-117, 2017.