广义Ramanujan-Nagell方程x^2-D=3~n的解数

The Number of Solutions of the Generalized Ramanujan-Nagell Equation x^2-D=3n

设D是不能被3整除的正整数.本文证明了:当D>1012时,如果Pell方程U2-DV2=-1有解(U,V),则方程x2-D=3n至多有2组正整数解(x,n).

Let D be a positive integer with D ≠ 0 （mod 3）. In this paper we prove that if D 〉 10^12 and the Pell equation U^2 - DV^2 = -1 has solutions （U, V）, then the equation x^2 - D = 3^n has at most two positive integer solutions （x, n）.