https://en.wikipedia.org/wiki/Formula_for_primes#Formula_bas...
I hadn’t heard of this result, and my exposure to Diophantine equations is limited to precisely one seminar from undergrad, but this feels like taking von Neumann’s famous quip to its most fantastical extreme:
> With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
"Via Gödel numbering, the theorems of an axiomatizable theory T become in effect an recursively enumerable set. The search for proofs becomes the search for solutions of a Diophantine equation. [...]
"Theorem. For any axiomatizable theory T and any proposition P, if P has a proof in T, then P has another proof consisting of 100 additions and multiplications of integers."