Wolfram Research

Function Repository Resource:

ProvablePrimeQ

Source Notebook

Certify a number as provably prime

Contributed by: Wolfram Research

ResourceFunction["ProvablePrimeQ"][n]

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

Details and Options

When ResourceFunction["ProvablePrimeQ"][n] returns True, then n is prime based on the Pratt certificate of primality or the Atkin–Morain certificate of primality.
ResourceFunction["ProvablePrimeQ"] should not be used as a replacement for PrimeQ, as PrimeQ is several orders of magnitude faster. Instead, use ResourceFunction["ProvablePrimeQ"] to certify the results of PrimeQ when needed.
The following options can be given:
"SmallPrime" 1050 lower bound for using the Atkin–Morain test
"Certificate" False whether to print a certificate
"PollardPTest" Automatic whether to use the Pollard p-1 method
"PollardRhoTest" Automatic whether to use the Pollard ρ method
"TrialDivisionLimit" Automatic number of primes to use in trial division
"PrimeQMessages" False whether progress is to be monitored

Examples

Basic Examples

PrimeQ indicates that 1093 is prime:

In[1]:=
PrimeQ[1093]
Out[1]=

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

In[2]:=
ResourceFunction["ProvablePrimeQ"][1093]
Out[2]=

Scope

ProvablePrimeQ works on arbitrarily large numbers:

In[3]:=
ResourceFunction["ProvablePrimeQ"][10^300 + 3]
Out[3]=

ProvablePrimeQ automatically threads over lists:

In[4]:=
ResourceFunction["ProvablePrimeQ"][{3, 7, 9, 13}]
Out[4]=

Options

Certificate

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

In[5]:=
ResourceFunction["ProvablePrimeQ"][1093, "Certificate" -> False]
Out[5]=

PrimeQMessages

A random prime:

In[6]:=
p = RandomPrime[{10^60, 10^61}]
Out[6]=

Progress messages are printed with "PrimeQMessages"True:

In[7]:=
ResourceFunction["ProvablePrimeQ"][p, "PrimeQMessages" -> True]
Out[7]=

Properties and Relations

Here is a random prime:

In[8]:=
p = RandomPrime[{10^9, 10^12}]
Out[8]=

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

In[9]:=
ResourceFunction["ProvablePrimeQ"][p]
Out[9]=
In[10]:=
ResourceFunction["PrimeQCertificate"][p]
Out[10]=

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

In[11]:=
ResourceFunction["ProvablePrimeQ"][p, "Certificate" -> True]
Out[11]=

Possible Issues

A certificate cannot be generated for -1, 0, or 1:

In[12]:=
ResourceFunction["ProvablePrimeQ"][0]
Out[12]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Resource History

See Also

License Information