certificate of compositeness

A compositeness certificate is a piece of information which guarantees that a given number p is composite. Possible certificates consist of a factor of a number (which, in general, is much quicker to check by direct division than to determine initially), or of the determination that either a^(p - 1) not congruent 1 (mod p), (i.e., p violates Fermat's little theorem), or a!=-1, 1 and a^2 congruent 1 (mod p). A quantity a satisfying either property is said to be a witness to p's compositeness.

Adleman-Pomerance-Rumely primality test | Fermat's little theorem | primality certificate | witness