TITLE: Can you solve the listed smallest open Diophantine equations?
QUESTION [48 upvotes]: In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and evaluate. For example, the size $H$ of the equation $y^2-x^3+3=0$ is $H=2^2+2^3+3=15$.
Below we will investigate only the solvability question: does a given equation has any integer solutions or not?
Selected trivial equations. The smallest equation is $0=0$ with $H=0$. If we ignore equations with no variables, the smallest equation is $x=0$ with $H=2$, while the smallest equations with no integer solutions are $x^2+1=0$ and $2x+1=0$ with $H=5$. These equations have no real solutions and no solutions modulo $2$, respectively. The smallest equation which has real solutions and solutions modulo every integer but still no integer solutions is $y(x^2+2)=1$ with $H=13$.
Well-known equations. The smallest not completely trivial equation is $y^2=x^3-3$ with $H=15$. But this is an example of Mordell equation $y^2=x^3+k$ which has been solved for all small $k$, and there is a general algorithm which solves it for any $k$. Below we will ignore all equations which belong to a well-known family of effectively solvable equations.
Selected solved equations.

The smallest equation neither completely trivial nor well-known is
$
y(x^2-y)=z^2+1$ with $H=17$. As noted by Victor Ostrik, it has no solutions because all odd prime factors of $z^2+1$ are $1$ modulo $4$.

The smallest equation not solvable by this method is
$
x^2 + y^2 - z^2 = xyz - 2
$
with $H=22$. This has been solved by Will Sawin and  Fedor Petrov  On Markoff-type diophantine equation by Vieta jumping technique.

The smallest equation that required a new idea was $y(x^3-y)=z^2+2$ with $H=26$. This one was solved by Will Sawin and Servaes by rewriting it as $(2y - x^3)^2 + (2z)^2 = (x^2-2)(x^4 + 2 x^2 + 4)$, see this comment for details.

Equation
$
y^2-xyz+z^2=x^3-5
$ with $H=29$ has been solved in the arxiv preprint Fruit Diophantine Equation (arXiv:2108.02640) after being popularized in this blog post.

Equation
$
x(x^2+y^2+1)=z^3-z+1
$
with $H=29$ has solution $x=4280795$, $y=4360815$, $z=5427173$, found by Andrew Booker. This is the smallest equation for which the smallest known solution has $\min(|x|,|y|,|z|)>10^6$.

Equation
$
x^3 + y^3 + z^3 + xyz = 5
$
with $H=37$ has been listed here as the smallest open symmetric equation, but then I found solution $x=-3028982$, $y=-3786648$, $z=3480565$, see the answer for details how it was found.


Smallest open equations. The current smallest open equation is
$$
y(x^3-y)=z^3+3.
$$
This equation has $H=31$, and is the only remaining open equation with $H\leq 31$. Also, the only open equations with $H \leq 32$ are this one and the two-variable ones listed below.
One may also study equations of special type. For example, the current smallest open equations in two variables are
$$
y^3+xy+x^4+4=0,
$$
$$
y^3+xy+x^4+x+2=0,
$$
$$
y^3+y=x^4+x+4
$$
and
$$
y^3-y=x^4-2x-2
$$
with $H=32$. The current smallest open cubic equations are listed in a separate question here. The current smallest open symmetric equation is
$$
x^3+x+y^3+y+z^3+z = x y z + 1
$$
with $H=39$, while the current smallest open 3-monomial equation is
$$
x^3y^2 = z^3 + 6
$$
with $H=46$.
The shortest open equations. I was told that it would be interesting to order equations by a more "natural" measure of size than $H$. Define the length of a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,...,a_k$ as $l(P)=\sum_{i=1}^k\log_2|a_i|+\sum_{i=1}^k d_i$. This is an approximation for the number of symbols used to write down $P$ if we write the coefficients in binary, do not use the power symbol, and do not count the operations symbols. Note that $2^{l(P)}=\prod_{i=1}^k\left(a_i2^{d_i}\right)$ while $H(P)=\sum_{i=1}^k\left(a_i2^{d_i}\right)$. If we order equations by $l$ instead of $H$, then the current "shortest" open equations are
$$
y(x^3-y) = z^4+1,
$$
$$
2 y^3 + x y + x^4 + 1 = 0
$$
and
$$
x^3 y^2 = z^4+2
$$
of length $l=10$.
For each of the listed equations, the question is whether they have any integer solutions, or at least a finite algorithm that can decide this in principle.
The paper Diophantine equations: a systematic approach devoted to this project is available online: (arXiv:2108.08705). Paper last updated 13.04.2022.
The plan is to list new smallest open equations once these ones are solved. The solved equations will be moved to the "solved" section.

REPLY [15 votes]: The equation
$$
x^3 + y^3 + z^3 + xyz = 5
$$
is solvable in integers. For example, take
$$
x=-3028982, \quad y=-3786648, \quad z=3480565.
$$
Verification is straightforward, but I would like to add more details how this solution has been found. If we just try $x$ and $y$ in order and then solve for $z$, then there are $10^{12}$ pairs $(x,y)$ even up to a million, hence finding the solution above was out of reach, at least for my computer.
Instead, I have noticed that linear change of variables
$$
y \to y - z, \quad x \to x + 3 y
$$
reduces the equation to
$$
-5 + x^3 + 9 x^2 y + 27 x y^2 + 28 y^3 + x y z - x z^2 = 0
$$
from which we can see that $28y^3-5$ is divisible by $x$. Hence, we may choose any $y$, then choose $x$ among the divisors of $28y^3-5$ and solve the resulting quadratic equation in $z$. This allows to find the above solution in just a few hours on standard PC.