Let us consider a real dynamical system $s′=g(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: For given fixed positive integers $a,b$, I am asking if this equation $$ x(2a-x)=b$$ has positive integer solutions $1≤x<a$.
We can find the solution:
$$x=a\pm\sqrt{a^2-b}$$
however, we need $x$ to be an integer for a given $a,b$. So, my question is about ideas that permit us to conclude that a positive integer solution $x$ exists. I notice that we do not need the concepts from mapping theory and this is just an algebraic equation.
Let $b=b_0b_1$ a factorization of $b$. If we have
$$x=b_0,\\2a-x=b_1$$ then $$2a=b_0+b_1.$$
So $a$ must be the arithmetic mean of two factors of $b$.