Abstract

Let <em>p</em> 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 <em>x</em> be an odd number, one necessary condition which the equation has integer solutions is that 2^{<em>n</em>(<em>p</em>-1)}-1/<em>p</em> contains some square factors. 2). Let <em>x</em> be an even number, when <em>n</em>=<em>pk</em>(<em>k</em>≥1), all integer solutions for the equation are (<span id="cke_bm_96S" style="display: none;"> </span><em>x</em><span id="cke_bm_96E" style="display: none;"> </span>, <em>y</em>)=(0,4^<em>k</em>); when <span id="cke_bm_106S" style="display: none;"> </span><span id="cke_bm_104S" style="display: none;"> </span><em>n</em><span id="cke_bm_106E" style="display: none;"> </span><span id="cke_bm_104E" style="display: none;"> </span>=<span id="cke_bm_107S" style="display: none;"> </span><em>pk</em><span id="cke_bm_107E" style="display: none;"> </span>+(<em>p</em>-1)<span id="cke_bm_108E" style="display: none;"> </span>/2(<span id="cke_bm_105E" style="display: none;"> </span><em>k</em>≥0), all integer solu<span id="cke_bm_97E" style="display: none;"> </span>tions are (±2^{<em>pk</em>+(<em>p</em>-1)}/2,2^{2<em>k</em>+1}); when <em>n</em>≡1,2,3,…,(p-3)/2, (<span id="cke_bm_124S" style="display: none;"> </span><em>p</em><span id="cke_bm_124E" style="display: none;"> </span>+1)/2,…,<em>p</em>-1(modp), the equati<span id="cke_bm_125E" style="display: none;"> </span>on has no integer solution.