Let be an odd prime, and let , be positive integers with .
For any nonnegative integer , let
and
,
where
and
. In this paper we prove the following results: (i) If , then
and the equation has only the positive integer solution
. (ii) If and
with
, then the equation has only the positive integer solution
.

Key Words: exponential diophantine equation; generalized Ramanujan-Nagell equation; Pell number.

2010 Mathematics Subject Classification: 11D61.

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