Q: Polynomial/ Exponential diophantine equation I am looking for the reference characterizing all the cases when 
$$an^2+bn+c=2^m$$
has infinitely many positive integer solutions (m,n). Thanks.

A: Consider:
(Ax+B)(Cx+D)=ACx^2 +(AD+BC)x + BD = ax^2 +bx + c , we have
a=AC
b=AD+BC
c= BD
For any arbitrary value of A and  x there is a number like B so that:   
B=2^m - Ax;   where m is an arbitrary number.
Similarly for any arbitrary value of C and x there is a number like D so that:
D=2^k - Cx;  where k is an arbitrary number.
Therefore There are infinitely many relations such as:
(Ax+B)(Cx+D)=2^m . 2^k = 2^(m+k) =2^n
That is the equation ax^2 +bx + c = 2^n has infinitely many solutions with above mentioned conditions. All factors and variable can be positive integers,
