PrimalityProving`
PrimalityProving`

PrimeQCertificate

PrimeQCertificate[n]

gives a certificate that n is prime or that n is composite.

Details and Options

• To use PrimeQCertificate, you first need to load the Primality Proving Package using Needs["PrimalityProving`"].
• PrimeQCertificate uses the Pratt certificate and the AtkinMorain certificate for primality.
• A certificate of compositeness is a list of three integers, either {a,n-1,n} or {a,2,n}, with .
• A prime always satisfies . The certificate {a,n-1,n} can be used to show that n is composite by demonstrating that .
• Any number whose square is for prime must satisfy . The certificate {a,2,n} 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 is a prime if one or more smaller numbers are prime as well.
• PrimeQCertificate has the same options as ProvablePrimeQ.

Examples

Basic Examples(2)

A certificate that can be used to prove that 1093 is a prime:

The same certificate can be obtained by using ProvablePrimeQ with the option "Certificate"->True:

A certificate that can be used to prove that 1093×3511 is composite:

The output is a list of three integers that indicate 1093×3511 is composite, and that it violates Fermat's little theorem for primes, if is prime: