广义Lebesgue-Nagell方程x2-4p2r＝y3

摘要

Let p be an odd prime ,using certain properties of the generalized Ramanujan-Nagell equations , the conclusion can be proved that the equation x2 -4 p2r= y3 have positive integer solutions (x ,y ,r) with gcd(x ,y)=1 if and only if p=3s2 +4 ,where s is an odd integer with s>1 .Moreover ,if the above condition holds ,then the equation has only the positive integer solution (x ,y ,r)=(s3 +12s ,s2 -4 ,1) with gcd(x ,y)=1 .%设 p是奇素数，运用广义Ramanujan-Nagell方程的性质证明了方程 x2-4 p2 r ＝ y3有适合gcd（x ，y）＝1的正整数解（x ，y ，r）的充要条件是 p ＝3s2＋4，其中s是大于1的奇数．当此条件成立时，该方程仅有正整数解（x ，y ，r）＝（s3＋12s ，s2-4，1）适合gcd（x ，y）＝1．