Hilbert $23$: Problem $23$
Further development of the calculus of variations.
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".