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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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    $\begingroup$ Concerning $x^3y^2=z^3+6$, once upon a time it was conjectured that no two powerful numbers differed by $6$. But then Narkiewicz found $7^325^2=463^2+6$. $\endgroup$ Jul 31, 2021 at 12:49
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    $\begingroup$ Unless I made an error, the first two equations and the symmetric one define smooth cubic surfaces. It would be interesting to calculate the Brauer group of the surfaces and determine whether there is a Brauer-Manin obstruction to solutions. If not, I suspect one can use techniques as in Heath-Brown's paper "The density of zeros of forms for which weak approximation fails" to guess the asymptotics for the number of solutions and make predictions about when one should be found. $\endgroup$
    – Will Sawin
    Jul 31, 2021 at 13:51
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    $\begingroup$ @Gottfried, since there haven't been any answers posted at all, I'd say it's a little premature to tag the question with big-list. $\endgroup$ Aug 16, 2021 at 13:13
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    $\begingroup$ If you look at the submission history, you will see that the paper has been submitted on the 2nd of August. So, equation (51) has been posed as an open problem in my paper submitted 2nd of August and then has been resolved in a later paper submitted 5th of August. It is a question to arxiv moderators why they needed 18 days to check that my paper is of sufficiently good quality to be accepted to the arXiv. $\endgroup$ Aug 20, 2021 at 12:37
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    $\begingroup$ @BogdanGrechuk A wordpress blog would work reasonably well: each equation could have its own blog post for discussion, as well as various tags ("H=46", "3-monomial", etc.) that one could use for navigation, and one could have additional pages that organize the equations in various ways (either updated by hand or by some automatic script). Alternatively one could look at databases such as the L-function and modular form database lmfdb.org for inspiration. $\endgroup$
    – Terry Tao
    Sep 16, 2021 at 14:52

1 Answer 1

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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.

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    $\begingroup$ Nice, +1! If I may ask, how did you notice that particular change of variables (or did you just try several simple linear changes)? $\endgroup$ Feb 17 at 18:51
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    $\begingroup$ I made substitution $y \to y-z$ to cancel $z^3$ and made the equation quadratic in $z$. The term near $z^2$ in this quadratic equation was $(3y-x)$, which suggested the second substitution. $\endgroup$ Feb 17 at 19:13
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    $\begingroup$ I have added new smallest open symmetric equation to the question: $x^3+x+y^3+y+z^3+z = x y z + 1$ with $H=39$. A similar transformation applies, which allows me in 24 hours check for solutions up to $|y+z|\leq 5,000,000$. No solutions found. $\endgroup$ Feb 19 at 8:31
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    $\begingroup$ After the new smallest open equation with $H=39$, the next symmetric equation the program returned has been $x^3+y^3+z^3-xyz=11$ with $H=43$. I used similar transformation, then did a lot of optimizations in the code, and then run it for a week. After a few days, the search returned the solution $x=-106179597, y=-530403428, z=567068662$. This is the first equation in this project such that in the smallest known solution all variables exceed $10^8$ in absolute value. In fact, I think such solutions are the largest ones that can be found by a direct search on one standard PC. $\endgroup$ Mar 9 at 9:48

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