# Primality Proving Package

This package implements primality proving. If

ProvablePrimeQ[n] returns

True, then the number

can be mathematically proven to be prime. In addition,

PrimeQCertificate[n] prints a certificate that can be used to verify that

is prime or composite.

Proving primality or compositeness.

The functions provided in this package not only prove primality, but they also generate a certificate of primality. A certificate of primality is a relatively short set of data that can be easily used to prove primality. The word easily means that using the data to prove primality is much easier and faster than generating the data in the first place. As a simple example of a certificate, the factors of a composite number provide a certificate of compositeness. Multiplying the numbers together to show that they are the factors is much easier than finding the factors. The advantage of providing certificates is that the user does not have to trust the internal mechanism of the algorithm that generated the certificate. It is fairly easy to write a program (in any system) that checks that the certificate provides a proof of primality.

PrimeQ indicates that the number 1093 is prime.

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

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This prints the certificate.

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This prints the certificate directly.

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The certificate of primality used in this package for large

n is based on the theory of elliptic curves.

This package can also be used to generate certificates of compositeness for composite numbers. These certificates are based on showing that simple properties that are true of prime numbers are not true for the given number. For example,

PrimeQCertificate returns the certificate

, which is intended to show that

is not equal to

.

ProvablePrimeQ returns

True or

False depending on whether

n is prime or not. The certificate for primality or compositeness is generated by

PrimeQCertificate.

ProvablePrimeQ calls

PrimeQCertificate and stores the result, so it does not take any extra time to create a certificate once

ProvablePrimeQ has returned an answer. The certificate generated by

PrimeQCertificate can be checked by

PrimeQCertificateCheck. This function recognizes whether the certificate asserts primality or compositeness and then uses the certificate to verify the assertion.

This prints the certificate of the prime number.

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This verifies primality using the certificate.

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Here is the certificate for a composite number. You can recognize a certificate of compositeness because it will always be a list of three integers.

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This verifies the compositeness.

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This package is not meant to replace the built-in primality tester

PrimeQ but rather to allow you to be completely secure that a number is truly prime. The package should be used only to certify results after all the number theoretic work has been done. For example, it would be a mistake to use

ProvablePrimeQ as a primality test for an integer-factoring algorithm. Rather, only when the complete factorization has been achieved (using

PrimeQ for primality testing) would you use

ProvablePrimeQ to certify the primality of the prime factors given by the algorithm. The reason for this is that

PrimeQ will be, in general, several orders of magnitude faster than

ProvablePrimeQ.

| | |

"SmallPrime" | 10^50 | lower bound for using the Atkin-Morain primality test |

"Certificate" | False | whether to print a certificate |

"PollardPTest" | Automatic | whether to apply the Pollard factoring algorithm in the search for the next prime in the recursive certificate |

"PollardRhoTest" | Automatic | whether to apply the Pollard factoring algorithm in the search for the next prime in the recursive certificate |

"TrialDivisionLimit" | Automatic | number of primes to be used in the trial division part of PrimeQCertificate |

"PrimeQMessages" | False | whether to print out the progress of the algorithm |

Options for ProvablePrimeQ and PrimeQCertificate.

When

is larger than the value of the option

,

ProvablePrimeQ uses the Atkin-Morain test as described above.

Note that you must use the same value of

when you check a certificate using

PrimeQCertificate that you used when you generated it using

ProvablePrimeQ or

PrimeQCertificate.

Since the default value of

is larger than the given number, Pratt's certificate of primality is returned.

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For large numbers, the certificate can be quite long and involved.

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