PrimalityProving`
PrimalityProving`

ProvablePrimeQ

ProvablePrimeQ[n]

gives True if n is provably prime, and False otherwise.

Details and Options

  • To use ProvablePrimeQ, you first need to load the Primality Proving Package using Needs["PrimalityProving`"].
  • When ProvablePrimeQ[n] returns True, then n is prime based on the Pratt certificate of primality or the Atkin–Morain certificate of primality.
  • ProvablePrimeQ should not be used as a replacement for PrimeQ, as PrimeQ is several orders of magnitude faster. Instead, use ProvablePrimeQ to certify the results of PrimeQ when needed.
  • The following options can be given:
  • "SmallPrime"1050lower bound for using the Atkin–Morain test
    "Certificate"Falsewhether to print a certificate
    "PollardPTest"Automaticwhether to use the Pollard method
    "PollardRhoTest"Automaticwhether to use the Pollard method
    "TrialDivisionLimit"Automaticnumber of primes to use in trial division
    "PrimeQMessages"Falsewhether progress is to be monitored

Examples

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Basic Examples  (1)

PrimeQ indicates that 1093 is prime:

ProvablePrimeQ gives the same result, but it has generated a certificate:

Scope  (2)

ProvablePrimeQ works on arbitrarily large numbers:

ProvablePrimeQ automatically threads over lists:

Options  (2)

Certificate  (1)

Use the option "Certificate"->True to view the certificate directly:

PrimeQMessages  (1)

A random prime:

Progress messages are printed with "PrimeQMessages"->True:

Properties & Relations  (1)

Here is a random prime:

If ProvablePrimeQ has returned a result, use PrimeQCertificate to print the certificate:

With "Certificate"->True, ProvablePrimeQ repeats the Atkin–Morain primality test:

Possible Issues  (1)

A certificate cannot be generated for , , or :