Linked Questions
12 questions linked to/from What is the smallest unsolved Diophantine equation?
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Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
72
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3
answers
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Can you solve the listed smallest open Diophantine equations?
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 ...
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5
votes
5
answers
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views
What are the integer solutions to $z^2-y^2z+x^3=0$?
The question is to describe ALL integer solutions to the equation in the title. Of course, polynomial parametrization of all solutions would be ideal, but answers in many other formats are possible. ...
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24
votes
2
answers
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On the smallest open Diophantine equations: beyond Hilbert's 10 problem
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
- 7,182
15
votes
1
answer
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Most wanted Diophantine equations
For most of my life, one single (family of) Diophantine equation(s) dominated the list of the world's most celebrated unsolved mathematical problems. Perhaps the world we live in now has grown too ...
8
votes
1
answer
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On Markoff-type diophantine equation
Do there exist integers $x,y,z$ such that
$$
x^2+y^2-z^2 = xyz -2 \quad ?
$$
Why this is interesting? First, this equation arose in an answer to the previous Mathoverflow question What is the smallest ...
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6
votes
1
answer
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views
$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
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4
votes
1
answer
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Are these equations solvable in positive integers?
By Matiyasevich theorem, there is no algorithm to decide whether a given Diophantine equation $P(x_1,\dots, x_n)=0$ has a solution in positive integers. As suggested in What is the smallest unsolved ...
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votes
1
answer
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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 ...
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0
votes
0
answers
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How to describe all integer solutions to $x^2+y^2=z^3+1$?
The question is to find all integer solutions to the equation
$$
x^2+y^2=z^3+1.
$$
This equation obviously has infinitely many integer solutions (take, for example, $(x,y,z)=(1,u^3,u^2)$ for any ...
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5
votes
3
answers
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Can you describe all rational solutions to these simple-looking equations?
Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
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7
votes
0
answers
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views
Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?
Given a Diophantine equation it is not decidable if it has integer solution.
I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties
every member in $\mathcal D_{unique}$ is a ...
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