广义Ramanujan-Nagell方程x~2+D~m=p~n

设D是大于1的正整数,p是不能整除D的素数.本文证明了:当D=3a~2+1,p=4a~2+1,其中a是正整数时,除了(D,p)=(4,5)这一情况以外,方程x~2+D~m=p~n仅有2组正整数解(x,m,n)=(a,1,1)和(8a~3+3a,1,3).根据上述结果得到了该方程解数的最佳上界.

On the Generalized Ramanujan-Nagell Equation x~2+D~m=p~n

Let D be a positive integer with D>1,and let p be a prime with p (?) D.In this paper we prove that if D=3a~2+1,and p=4a~2+1,where a is a positive integer, then the equation x~2+D~m=p~n has only two positive integer solutions (x,m,n) = (a,1,1) and (8a~3+3a,1,3),except for (D,p)=(4,5).By the above mentioned result, the best upper bound for the number of solutions of this equation is given.