Fait mathématique: 5278 is the number of ways, up to symmetry, to pick 3 elements of an 8×8 grid.
Culture générale: 5278 is a number for which we're missing a fact (submit one to numbersapi at google mail!).
Date historique: June 13th is the day in 1525 that Martin Luther marries Katharina von Bora, against the celibacy rule decreed by the Roman Catholic Church for priests and nuns.