Constraint diophantine equations (open problem)

Question

Given an odd number $N$ such that $N\equiv 3 \mod 4$, I am looking to find numerically or algebraically a positive natural number $k$ s.t, $1\leqslant k \leqslant \frac{N-3}{4}$, satisfying the quadratic equation: \begin{equation}\label{e1} (a+1)^2=2(N+1)-4k-b \end{equation} Where $a$ & $b$ are defined by $$a=\frac{N-3}{4}-3(k+1)+m$$ $$b=16k^2+(40-4m)k-m(N+5)+24$$ And $$m=\left\lfloor\frac{(4k+5)^2-1}{4k+5+N}\right\rfloor$$ under the constraint which are $$5k-m=\left\lceil\frac{(5k-m)^2+4kN-\frac{N^2-30N+609}{16}}{10k-2m+\frac{N+25}{2}}\right\rceil , \quad \mbox{prime factors of $N$ are unknown}$$ Where $\lfloor x \rfloor$ and $\lceil x \rceil$ denotes respectively the floor function and the ceil function of the real number $x$. If the given odd number $N$ is small enough, I can find the solution to this problem without knowing its prime factors. The problem is when $N$ is taken big enough. The brute force algorithm is usless, and factorization can take ages!!