Please check my post.
It indicates that the number of divisors of $x^2+x+41$ is equal to the number of lattice points of $X^2+163Y^2-2(2x+1)Y-1=0$.
This formula is transformed in this way.
$$163X^2+163^2Y^2-2\cdot163(2x+1)Y=163$$ $$163X^2+\{163Y-(2x+1)\}^2-(2x+1)^2=163$$ $$163X^2+(163Y-2x-1)^2=4x^2+4x+164$$
$X':=163Y-2x-1,\ Y':=X$ and we divide both sides by 4,
$$\frac{{X'}^2+{163Y'}^2}{4}=x^2+x+41=P(x)$$
$$N\left(\frac{X'+Y'\sqrt{-163}}{2}\right)=P(x)$$
$\frac{X'+Y'\sqrt{-163}}{2}$ is an element of $\mathbb Q(\sqrt{-163})$.
This formula indicates that the elements of $\mathbb Q(\sqrt{-163})$ with norm $P(x)$ is linked to the product pattern of two integers of $P(x)$.
The phenomenon you are interested in is based on this fact.
For example, let $x$ be $76$.
\begin{eqnarray*} P(x)&=&76^2+76+41\\ &=&1\cdot5893\\ &=&71\cdot83 \end{eqnarray*}
The number of product pattern of $P(x)$ is $2$.
On the other hand, the elements of $\mathbb Q(\sqrt{-163})$ with norm $P(x)$ (ignore sign) are
$$\frac{153+\sqrt{-163}}{2},\ 5+6\sqrt{-163}\ .$$
The number is $2$.
The left one is a trivial element $\frac{(2x+1)+\sqrt{-163}}{2}$.The other one is a non-trivial element that indicates that $P(x)$ is a composite number.