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 equation is the equation
$$
3-y+x^2 y+y^2+x y z-2 z^2 = 0,
$$
of size $H=33$, see 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.

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