On The Generalized Ramanujan–Nagell Equation x2+(2c-1)m=cn

We show that if c is a positive integer satisfying 2 c - 1 = 3 p l or 2 c - 1 = 5 p l with p prime and l positive integer, then the equation x 2 + ( 2 c - 1 ) m = c n has only the positive integer solution ( x , m , n ) = ( c - 1 , 1 , 2 ) without any congruence condition on a prime p .