Fermat's Last Theorem was open for more than 350 years until Andrew Wiles proved it in 1995. Are there possible (historical or other) reasons why David Hilbert did not include this famous open problem in his problem list in 1900?
Hilbert included in his list a much more general Problem 10:
Given a Diophantine equation with any number of unknown quantities and with rational integral 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.
Probably he did not think that Fermat's equation by itself is sufficiently important to deserve a separate entry in his list.