Search epoch.ai
Finiteness Problem for Diophantine Equations
Prove that certain “small” Diophantine equations have infinitely many solutions.
On this page:
About the problem
Attempts by AI
AI prompts
Mathematician survey
About the problem
Define a heuristic “size” of a Diophantine equation by substituting $2$ for all variables and absolute values of coefficients for all coefficients. For instance $x^2 - 2y^2 -1$ has size $13$.
The finiteness problem for a Diophantine equation asks whether the equation has infinitely many solutions. This problem is about the nine equations with size $\leq 24$ where the finiteness problem is open. Numerical evidence suggests that these equations all have infinitely many solutions, but this has not been proven.
Each of these equations has been attempted by the problem authors and others on MathOverflow. The authors believe there is a high chance that a new, interesting method will be required to solve at least some of these equations, even if it turns out that some equations have relatively uninteresting resolutions. The problem is conservatively marked as “moderately interesting”, and only considered solved when all nine equations are solved. In hindsight, it may be clear that even a partial solution is interesting.
The authors note their uncertainty about timeframe: “You never know with Diophantine equations — who could initially predict that Fermat’s Last Theorem would take centuries to prove?”
Attempts by AI
We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.
AI Prompts
The full problem prompt should be posed independently nine times, filling in the blank with each of the following equations each time: z^2+y^2z+x^3-2=0, z^2+y^2z+x^3-x-1=0, z^2+y^2z+x^3+x-1=0, z^2+y^2z+x^3+x+1=0, z^2+y^2z+x^3-3=0, z^2+y^2z+x^3+3=0, z^2+y^2z+x^3-x-2=0, z^2+y^2z+x^3-x+2=0, z^2+y^2z-z+x^3+2=0
Warm-up
Full problem
Mathematician survey
The author assessed the problem as follows.