Hilbert $23$: Problem $10$
Hilbert's $10$th problem was the question:
- Given a Diophantine equation with any number of unknown quantities and with rational integer numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
The MRDP Theorem has shown that recursively enumerable sets are equivalent to Diophantine sets.
Matiyasevich's Theorem shows that solutions to Diophantine equations may grow exponentially.
The Hilbert 23 were delivered by David Hilbert in a famous address at Paris in $1900$.
He considered them to be the oustanding challenges to mathematicians in the future.
There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".
- (translated by Mary Winston Newson from "Mathematische Probleme")