广义Lebesgue-Nagell方程x^2-4p^(2r)=y^3

The generalized Lebesque-Nagell equation x^2-4p^(2r)=y^3

设p是奇素数,运用广义Ramanujan-Nagell方程的性质证明了方程x2-4p2r=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.

Let p be an odd prime,using certain properties of the generalized Ramanujan-Nagell equations, the conclusion can be proved that the equation x^2-4p^2r=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）l. Moreover, if the above condition holds, then the equation has only the positive integer solution （x,y,r）= （sa ＋12s,s^2 -4,1） with gcd（x,y）= 1.