OddQ[expr] gives True if expr is an odd integer, and False otherwise.
OddQ[expr] returns False unless expr is manifestly an odd integer (i.e., has head Integer, and is odd).
You can use OddQ[x] ^= True to override the normal operation of OddQ, and effectively define x to be an odd integer.
See The Mathematica Book: Section 2.3.5 and Section 3.1.1.
See also: IntegerQ, EvenQ, TrueQ.