Timeline for What is the smallest unsolved Diophantine equation?
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| Dec 19, 2023 at 10:07 | comment | added | Bogdan Grechuk | Quartic equations up to $17$ are easy to list, so consider at most cubic. Assume we have $n=4$ variables, say $x,y,z,t$. List all possible monomials of degree $d\leq 3$ without coefficients, that is, $1,x,y,z,t,x^2,y^2,xy,...$. Consider all combinations of such monomials with total $h$ up to $17$. For some combinations, for example $1,x^3,yz,t$, all equations are easy no matter what are coefficients. We can exclude such combinations. For each of the remaining combinations, consider all possible coefficients. Then do the same for $n=2,3,5,...$ variables. On my computer, this works quite fast. | |
| Dec 19, 2023 at 0:55 | comment | added | TheOutZ | Grechuck After pondering it a bit more and coding it up, I admit my defeat. While I can see how eliminating "easy equations" might reduce the search space a tiny bit, I fail to see how we get all (or quickly assure of trivialness of most) of these equations in the first place - after all, the space of possibilities grows (even with a modest bound on the coefficients) enormously fast. Let's phrase it this way: How did you, for example, manage to find all equations up to $h=17$ in the original question (ignoring verifying solutions for a moment)? Am I just too blind to see it? | |
| Dec 13, 2023 at 12:58 | comment | added | Bogdan Grechuk | There is no magic here, just a step-by-step work. We start with a naïve way, probably one of a few you discovered. Then we eliminate easy equations. For example, all equations of the form ax_i = P where P is a polynomial in variables other than x_i are easy. And so on. With no restrictions on the equations and the problem, the smallest open ones have size H=17. For restricted problems we need to go deeper but can do search more efficiently. For example, if we only need to determine whether solutions exist, we can eliminate all equations with small solutions, which greatly reduces the search. | |
| Dec 12, 2023 at 23:37 | comment | added | TheOutZ | Excuse my late curiosity, but I have a small question: In your answers, you mentioned an algorithm that computes you all diophantine equations of a certain height. While I have found a few naïve ways to do just that, I can not for the life of me figure out how to make this computationally feasible for larger $h$. Do you mind sharing how you have done this? | |
| Feb 5, 2022 at 9:16 | history | edited | Matthieu Romagny | CC BY-SA 4.0 |
added 1 character in body
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| Nov 5, 2021 at 13:09 | history | answered | Bogdan Grechuk | CC BY-SA 4.0 |