Find all integer solutions to the following easy-looking Diophantine equations

In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation $$P(x_1,\dots,x_n)=0$$ is solved if we have proved that its integer solution set $$S \subset {\mathbb Z}^n$$ can be represented as the finite union $$S=S_1 \cup \dots \cup S_m$$, where each $$S_i$$ is either a polynomial family or a family defined by recurrence relations. Here, $$S \subset {\mathbb Z}^n$$ is a polynomial family if there exists polynomials $$P_1,\dots,P_n$$ in some variables $$u_1,\dots,u_k$$ and integer coefficients such that $$(x_1,\dots,x_n) \in S$$ if and only if there exists integers $$u_1,\dots,u_k$$ such that $$x_i=P_i(u_1,\dots,u_k)$$ for $$i=1,\dots,n$$.

Following Zidane What is the smallest unsolved diophantine equation? , let us define size $$H$$ of the equation $$P=0$$ as a result of substitution 2 instead of all variables, absolute values instead of all coefficients, and evaluating.

All equations with $$H \leq 8$$ are easy to solve. However, simple-looking equation $$xy-zt=1$$ with $$H=9$$ has been open for decades. In 2010, Vaserstein https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-s.pdf proved that

• The solution set to $$xy-zt=1$$ is a polynomial family with $$46$$ parameters.

As a corollary of this, Vaserstein solved the following families of equations

• $$xy-zt=D$$ for any integer $$D$$;
• $$yz=x^2+D$$ for any integer $$D$$;
• $$x_1x_2+x_3x_4+Q(x_5,\dots,x_n)=D$$ for quadratic form $$Q$$ and integer $$D$$.

In addition, the following equations/families has been solved:

• Equations in the form $$dyz=ax^2+bx+c$$ for integers $$a,b,c,d$$ has been solved in the answer to question Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$
• All equations with $$H \leq 12$$.

I was also able to solve all equations with $$H=13$$ except of the following ones. $$x^3 + 1 = yz$$ $$x^2y=z^2 \pm 1$$ $$x^2y=tz+1$$ $$x^2 + y^2 \pm 1 = zt$$ $$x^2 \pm 1 = yzt$$ $$x_1x_2x_3+x_4x_5=1$$

For each of the listed equations, the question is to find all integer solutions. Specifically, check whether the set of all integer solutions is a finite union of polynomial families and/or families defined by recurrence relations. You do not need to write the resulting families explicitly, because, as example $$xy-zt=1$$ indicates, they may be quite complicated.

See here Can you solve the listed smallest open Diophantine equations? for a version of this question where we only want to check whether any integer solution exists, and here On the smallest open Diophantine equations: beyond Hilbert's 10 problem for a version where we also check whether the solution set is finite or infinite (and find all solutions if there are finitely many).

The equation $$x^2+1 = yzt$$ has a parametric solution as follows.

We factorize $$x^2+1 = yzt$$ in $$\mathbb Z[i].$$

Let $$(Y,Z,T)=(a+bi,c+di,e+fi)$$ then

$$x+i = YZT = (acf+ead+ebc-bdf)i+ace-fad-fbc-bde.$$

Hence we get a parametric solution $$x = acf+ead+ebc-bdf$$ if $$ace-fad-fbc-bde =1.$$

• We are not allowed to use the "if" condition, otherwise there is a trivial answer $x=a$, $y=b$, $z=c$, $t=d$ if $a^2+1=bcd$. We can only use "if" condition if we know how to solve the equation under if statement. So we have reduced the problem of writing parametric solutions of $x^2+1=yzt$ to the problem of writing parametric solutions of $ace-fad-fbc-bde=1$. This equation does not look any easier for me than the original one. Jan 20, 2022 at 7:42
• This condition is necessary to obtain all integer solutions, just as we did for $yz=x^2+x+1$. Jan 20, 2022 at 9:53
• With $yz=x^2+x+1$ we used the condition $ab-cd=1$ which we were allowed to use only because this equation has been solved in the Vaserstein paper. He proved that $ab-cd=1$ if and only if $a=P(u)$, $b=Q(u)$, $c=R(u)$, $d=T(u)$ where $P,Q,S,T$ are some polynomials and $u$ is a vector of variables without any conditions. Jan 20, 2022 at 14:05
• Then is we have $x=ad+bc+bd$, $y=a^2+ab+b^2$, $z=c^2+cd+d^2$, then $x=P(u)T(u)+Q(u)R(u)+Q(u)T(u)$, $y=P^2(u)+P(u)Q(u)+Q^2(u)$, $z=S^2(u)+S(u)T(u)+T^2(u)$ gives the final answer without any conditions on parameters $u$. This is not available for the condition $ace-fad-fbc-bde=1$. In general, we may use any condition defined by an equation that we have already solved (or that is solved in the literature). Jan 20, 2022 at 14:16
• Okay, I hope it will be parameterized. Jan 21, 2022 at 0:56