In 1937 Vinogradov proved that this is true for sufficiently large odd integers n. In 1956 Borodzkin showed n > 314348907 is sufficient (the exponent is 315). In 1989 Chen and Wang reduced this bound to 1043000.www.ddhw.com
Vinogradov, I. M. "Representation of an Odd Number as a Sum of Three Primes." Comptes rendus (Doklady) de l'Académie des Sciences de l'U.R.S.S.15, 169-172, 1937.