primeclassify

A library of functions to classify prime numbers

Each classification function takes an argument, p, which is assumed to be a prime number. There also are optional arguments tout and store. tout is the maximum processing time the function may take and store is the maximum number of data values it can store. These are self enforced. Some have additional optional arguments.

Generally, classifications that depend on a number base assume base 10.

Each returns one of:

• False (p not a member of that classification), or
• None (could not complete test), or
• something other than False or None (p a member of that classification) — usually True, but sometimes additional information (such as, for twin primes, which primes it is a twin of)

A return value of None may mean it is literally unknown whether p is in that classification, or just that the function doesn't know and doesn't care to find out. For example, the largest known Mersenne prime is M_82,589,933 = 2^82,589,933 − 1, but the mer function will return 'None' for any 'p' larger than M₁₂₇.

There are in addition a few utility functions.

Example usage

from primeclassify import pc
import sympy

x = 137
# pc functions do not check for primality, do that first
if not sympy.isprime(x):
    print (f"{x} is not prime")
# Test if x is a Chen prime
# Limit time to 0.01 seconds, storage to 1000 numbers
if pc.chen(x, tout=.01, stor=1000):
    print (f"{x} is a Chen prime")

Output is

137 is a Chen prime

Classification functions

• balanced (p, tout=0, stor=0) Balanced prime
• chen (p, tout=0, stor=0) Chen prime
• circular (p, tout=0, stor=0) Circular prime
• cluster(p, tout=0, stor=0) Cluster prime
• cousin(p, tout=0, stor=0) Cousin prime (returns False or tuple of its cousins)
• cuban(p, tout=0, stor=0) Cuban prime (returns False or (1 or 2 indicating which series))
• cullen(p, tout=0, stor=0) Cullen prime
• delicate(p, tout=0, stor=0) Delicate prime
• dihedral(p, tout=0, stor=0) Dihedral prime
• dbmer(p, tout=0, stor=0) Double Mersenne prime
• emirp(p, tout=0, stor=0) Emirp
• even(p, tout=0, stor=0) Even prime
• factorial(p, tout=0, stor=0) Factorial prime
• fermat(p, tout=0, stor=0) Fermat prime
• fibo(p, tout=0, stor=0) Fibonacci prime
• fortunate(p, tout=0, stor=0) Fortunate prime
• good(p, tout=0, stor=0) Good prime
• happy(p, tout=0, stor=0) Happy prime
• higgs(p, expt=2, tout=0, stor=0) Higgs prime with exponent expt
• lartrunc(p, tout=0, stor=0) Left-and-right-truncatable prime
• ltrunc(p, tout=0, stor=0) Left-truncatable prime
• lucas(p, tout=0, stor=0) Lucas prime
• mer(p, tout=0, stor=0) Mersenne prime
• mills(p, tout=0, stor=0) Mills prime
• minimal(p, tout=0, stor=0) Minimal prime
• motzkin(p, tout=0, stor=0) Motzkin prime
• nsw(p, tout=0, stor=0) Newman–Shanks–Williams prime
• pal(p, tout=0, stor=0) Palindromic prime
• pell(p, tout=0, stor=0) Pell prime
• pelllucas(p, tout=0, stor=0) Pell-Lucas prime
• permutable(p, tout=0, stor=0) Permutable prime
• pierpont(p, tout=0, stor=0) Pierpont prime
• pillai(p, tout=0, stor=0) Pillai prime
• primequadruplet(p, tout=0, stor=0) Prime quadruplet prime (returns False or tuple of triples of other three members of quadruplets)
• primetriplet(p, tout=0, stor=0) Prime triplet prime (returns False or tuple of duples of other two members of triplets)
• primorial(p, tout=0, stor=0) Primorial prime
• proth(p, tout=0, stor=0) Proth prime
• pyth(p, tout=0, stor=0) Pythagorean prime
• quartan(p, tout=0, stor=0) Quartan prime
• repu(p, tout=0, stor=0) Repunit prime
• rtrunc(p, tout=0, stor=0) Right-truncatable prime
• safe(p, tout=0, stor=0) Safe prime
• sexy(p, tout=0, stor=0) Sexy prime (returns False or tuple of its sexy partners. Yes, I said that.)
• sophie(p, tout=0, stor=0) Sophie Germain prime
• strobe(p, tout=0, stor=0) Strobogrammatic prime
• strong(p, tout=0, stor=0) Strong prime
• superprime(p, tout=0, stor=0) Super-prime
• supersing(p, tout=0, stor=0) Supersingular prime (of moonshine theory)
• twin(p, tout=0, stor=0) Twin prime (returns False or tuple of its twins.)
• wagstaff(p, tout=0, stor=0) Wagstaff prime
• wief(p, tout=0, stor=0) Wieferich prime
• williams(p, b=3, tout=0, stor=0) Williams prime (returns False or n)
• wilson(p, tout=0, stor=0) Wilson prime
• wolsten(p, tout=0, stor=0) Wolstenhome prime
• woodall(p, tout=0, stor=0) Woodall prime

Utility functions

• class_from_list (p, thelist, complete, limit=None) Used internally
• describe(p, tout=0, stor=0, extras={higgs: (2,), williams: (3, 10)}) Returns a list of classifications passed by p. Functions are called with tout=tout, stor=stor. extras give limits for additional arguments.
• test_classify (fn, limit1, limit2=-1, tout=0, stor=0, extra=None) Calls function fn for primes p in range [2, limit1] or [limit1, limit2], with tout=tout, stor=stor, and prints results. extra is extra argument for functions that take one.

Classifications not included

These are not integer primes:

• Eisenstein prime
• Gaussian prime

These are just too hard for me to code!

• Full reptend prime
• Genocchi prime
• Highly cototient prime
• Lucky prime
• Ramanujan prime
• Regular (and irregular) prime
• Supersingular prime (of algebraic number theory)
• Wall-Sun-Sun prime

Author: Rich Holmes
Source repository: https://gitlab.com/rsholmes/primeclassify