PRIMALITY PROVING PACKAGE SYMBOL

# PrimeQCertificate

PrimeQCertificate[n]
gives a certificate that n is prime or that n is composite.

## DetailsDetails

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

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

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

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

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A certificate that can be used to prove that is composite:

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

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