A general dictionary would define primality as “the quality or condition of being a prime number.” In mathematics, it might be more useful to define primality as a Boolean-valued function that returns True if the input number is prime and False otherwise. Two examples: the primality of 47 is True; the primality of 42 is False.

It is not necessary to perform integer factorization to know the primality of a given integer, as there are various congruences and other relations which prime numbers satisfy but non-primes don't; these can serve as primality tests. The primality of certain large numbers, such as the thirtieth Fermat number, has been determined even though all we know of its least prime factor is that it is less than the square root of the composite Fermat number. Before the primality of a large number is ascertained, it might be considered a probable prime. 1 is the only integer to be declared non-prime without a previously unknown factor being discovered.