Mathematical fact: 344 is the smallest number that can be written as the sum of a cube and a 7^{th} power in more than one way.
Trivia: 344 is a number for which we're missing a fact (submit one to numbersapi at google mail!).
Historical date: December 9th is the day in 1946 that the "Subsequent Nuremberg Trials" begin with the "Doctors' Trial", prosecuting doctors alleged to be involved in human experimentation.