  From the results in the literature, the algebraic set of the hyperbola with
parameter $n$ defined by $\mathcal{B}_{n}(X, Y, Z)_{\mid_{x\geq 4n}}=
\displaystyle \lbrace \left(X: Y: Z\right)\in \mathbb{P}^{2}(\mathbb{Q}) \
\vert \ \displaystyle Y^{2}=X^{2}-4nXZ \rbrace$ where $n$ is a semiprime is
proved to be in relation with prime factors of $n$. In the affine space over
$\mathbb{Z}_{\geqslant 4n}\times \mathbb{Z}_{\geqslant 0}$, this set has
exactly 5 points $\displaystyle\lbrace P_{0}, P_{1}, P_{2}, P_{3}, P_{4}
\rbrace$ with $P_{2}+P_{3}=P_{1}+2P_{2}=P_{4}$ for which knowledge of $P_{2}$
or $P_{3}$ yields the factorization of $n$. However, The non cyclicity of this
group structure over rationals and integers and moreover its non good reduction
over finite fields constitute the main difficulty in finding its solutions. In
this paper we describe an approach to finding $P_{2}$ and $P_{3}$. We introduce
the concept of Hyperbola X-root and Y-root that the solution's greatest common
divisors with $n$ reveal prime factors of $n$. We prove that $P_{2}$ and
$P_{3}$ can be found on a singular Weierstrass curve isomorphic to a Jacobi
quartic using the Hyperbola X-root and Y-root. We present the mathematical
framework for this approach.
