This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 PRIMALITY PROVING PACKAGE SYMBOL

# PrimeQCertificate

 PrimeQCertificate[n] gives a certificate that n is prime or that n is composite.
• uses the Pratt certificate and the Atkin-Morain certificate for primality.
• A certificate of compositeness is a list of 3 integers, either or , with .
• A prime p always satisfies . The certificate can be used to show that n is composite by demonstrating that .
• Any number a whose square is for n prime must satisfy . The certificate can be used to show that n is composite by demonstrating that and .
• A certificate of primality consists of a recursive list of certificates which prove that n is a prime if one or more smaller numbers are prime as well.
A certificate that can be used to prove that 1093 is a prime:
The same certificate can be obtained by using with the option True:
A certificate that can be used to prove that is composite:
The output is a list of 3 integers that indicate is composite, and that it violates Fermat's little theorem for primes, if p is prime:
Needs["PrimalityProving`"]
A certificate that can be used to prove that 1093 is a prime:
 Out[2]=
The same certificate can be obtained by using with the option True:
 Out[3]=

Needs["PrimalityProving`"]
A certificate that can be used to prove that is composite:
 Out[2]=
The output is a list of 3 integers that indicate is composite, and that it violates Fermat's little theorem for primes, if p is prime:
 Out[3]=