<s>[INST] Are there infinitely many positive integer solutions to $(xz+1)(yz+1)=P(z)$? Let $P(z)\equiv 1($ mod $ \ z) $ be a polynomial of degree $n>3$ with integer coefficients. Are there infinitely many positive integers $x, y, z$ such that $(xz+1)(yz+1)=P(z)$?
If $P(z) = a_nz^n+1$, it has be proven that the Diophantine equation has infinitely many solutions in positive integers $x, y, z$ Prove that the diophantine equation $(xz+1)(yz+1)=az^{k}+1$ has infinitely many solutions in positive integers..
From experiment, it appears the assertion is true for all polynomials $P(z)\equiv 1$(mod$ \ z)$ of degree $n>3$. How do we go about proving this?
Note if $n=3$, it has been shown that the Diophantine equation has a finite number of solutions
https://mathoverflow.net/questions/392002/is-xz1-a-proper-divisor-of-a-3z3a-2z2a-1z1-finitely-often/392018#392018
 [/INST] I figured out all solutions in positive integers $\ x, y , z \ $  to $(xz+1)(yz+1)=z^4+z^3+z^2+z+1$.
First, define some sequences as follows:
$r_m = m^2+m-1, \ \ \ m=1, 2, \ldots $
$q_m = (r_m+2)^2-2,\ \ \ m=1, 2, \ldots $
$a_m = m+1, \ \ \ m=1,2,\ldots $
$b_m = m^5 + 3m^4 + 5m^3 + 4m^2 +m, \ \ \ m=1,2,\ldots $
$c_m = m^3 + 2m^2 +2m, \ \ \ m=1,2,\ldots $
$e_m = m^3 + m^2 +m+1, \ \ \ m=1,2,\ldots $
$g_m = m^5 + 2m^4 + 3m^3 + 3m^2 +m, \ \ \ m=1,2,\ldots $
And for a particular $m$, define sequences $A,B,C,E,F,G$ as follows;
$A_1 = a_m$,  $A_2 = b_m$, $A_n = q_mA_{n-1} - A_{n-2} - r_m, \ \ \ n = 3, 4 , \dots$
$B_n = A_{n+1}, \ \ \ n = 1, 2, \ldots $
$C_1 = c_m$,  $C_2 = q_mC_1+r_m$, $C_n = q_mC_{n-1} - C_{n-2} + r_m, \ \ \ n = 3, 4 , \dots$
$E_1 = e_m$,  $E_2 = q_mE_1-r_m$, $E_n = q_mE_{n-1} - E_{n-2} - r_m, \ \ \ n = 3, 4 , \dots$
$F_n = E_{n+1}, \ \ \ n = 1, 2, \ldots $
$G_1 = g_m$,  $G_2 = q_mG_1+ r_m - m$, $G_n = q_mG_{n-1} - G_{n-2} + r_m, \ \ \ n = 3, 4 , \dots$
All positive integer solutions $(x,y,z)$ are given by $(A_n, B_n, C_n)$, $(E_n, F_n, G_n), n = 1, 2, \dots $.
NB. This is simply an observation. A proof is required to show that $(A_n, B_n, C_n)$, $(E_n, F_n, G_n)$ are indeed solutions for every $n$ and that these are the only solutions in positive integers.
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