A few questions regarding a polynomial of degree 2:
$f(x) = ax^2 + bx - c$
where $a,b,c$ are given positive powers of 2.
Or more explicitly:
$f(x) = 2^\alpha x^2+2^\beta x - 2^\gamma$
where $\alpha,\beta,\gamma$ are positive integers
1) Can one theoretically (I mean not by an algorithm which include any trial and error parts) determine if there is a positive integer value $x_0$ such that $f(x_0)$ is a square (namely that there is a positive integer $y$ such that $f(x_0)=y^2$) depending on the values of $a,b,c$?
2) Can one generate all solutions given one solution?
3) Given there is a solution, is there an algorithm to yield the solution?
Thanks
If so we can write $$2^{\alpha}x^2+2^{\beta}x-2^{\gamma}=(Ax+B)^2$$ so we get $$2^{\alpha}=A^2$$ $$2AB=2^{\beta}$$ $$-2^{\gamma}=B^2$$ The last equation isn't fulfilled for any real number $$\gamma,B$$