<s>[INST] Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$ Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set to the diophantine equation in positive integers $(x, y, z)$.
First,  define functions $r$ and $q$ of a positive integer $m$ as follows:
\begin{align} 
r(m) &= m^2+m-1 \\
q(m) &= (r(m) +2)^2-2 \\
\end{align}
And for a particular positive integer $m$, define sequences $A, B, C, D$ as follows;
\begin{align}
A_1 &= m+1\\
A_2 &= m^5 + 3m^4 + 5m^3 + 4m^2 +m\\
A_n &= q(m)A_{n-1} - A_{n-2} - r(m)\\
\end{align}
\begin{align}
B_1 &=  m^3 + 2m^2 +2m\\
B_2 &= q(m)B_1+r(m)\\
B_n &= q(m)B_{n-1} - B_{n-2} + r(m) \\
\end{align}
\begin{align}
C_1 &=  m^3 + m^2 +m+1 \\
C_2 &= q(m)C_1-r(m)\\
C_n &= q(m)C_{n-1} - C_{n-2} - r(m)\\
\end{align}
\begin{align}
D_1 &=  m^5 + 2m^4 + 3m^3 + 3m^2 +m\\
D_2 &= q(m)D_1+ r(m)- m\\
D_n &= q(m)D_{n-1} - D_{n-2} + r(m) \\
\end{align}
It appears all positive integer solutions $(x,y,z)$ are given by $(A_n, A_{n+1} , B_n)$, $(C_n, C_{n+1}, D_n), n = 1, 2, \dots $.
How does one go about proving that  $(A_n, A_{n+1} , B_n)$, $(C_n, C_{n+1}, D_n)$ are indeed solutions for all $n$ and showing that these are the only solutions? My approach would be to obtain closed formulae for $A_n, B_n, C_n$ and $ D_n$ then substitute into $(xz+1)(yz+1)=P(z)$ and check if the $LHS=RHS$. However, the closed forms of these sequences are too bulky. And also this method does not prove whether these are the only solutions.
If sequences $A, B, C, D $ cover all solutions in positive integers $x, y, z$ then it becomes much more efficient to determine if $P(z) $ possesses any proper factor of the form $xz+1$ for a  particular $z$ using these sequences as compared to trial divisions on $P(z) $ or prime factorizing $P(z) $.
Also, is it possible to construct all solutions for irreducible polynomials $P(z) = z^n+z^{n-1}+ \cdots + z^2+z+1$ of degree $n>4?$
 [/INST] "OP" wants to know of a numerical solution which does not satisfy his parametric solution. It is given below.
$(x,y,z)=(138,49331482518,2609165)$
</s>