Polynomial/ Exponential diophantine equation

Question

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.

Answer 1

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,