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$\begingroup$ More generally, there are many conjectures of the form $\forall n : \Phi(n)$ where $\Phi(n)$ is a statement we can confirm computationally for a given $n$ if it is true, but don't know how to disconfirm if it is false. E.g., we can conjecture that $n$ is a sum of three cubes if and only if $n\not\equiv \pm 4 \bmod 9$. The question "is $n$ a sum of three cubes?" will be trivially decidable if the conjecture is true, but we currently have no way to show that a given $n\not\equiv \pm 4 \bmod 9$ is not the sum of three cubes. Conceivably, the conjecture is false and the question is undecidable. $\endgroup$

Timothy Chow
– Timothy Chow

2025-11-21 15:35:44 +00:00
Commented Nov 21, 2025 at 15:35

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