imp/celeste/math/primes.py

from math import gcd, inf

from celeste.math.numbers import bigomega, factors
from celeste.extern.primefac import (
    isprime,
    primegen as Primes,
)

def coprime(n: int, m: int) -> bool:
    return gcd(n, m) == 1

def almostprime(n: int, k: int) -> bool:
    '''
    A natural n is "k-almost prime" if it has exactly
    k prime factors (including multiplicity).
    '''
    return (bigomega(n) == k)

def semiprime(n: int) -> bool:
    '''
    A semiprime number is one that is 2-almost prime.
    Ref: https://en.wikipedia.org/wiki/Semiprime 
    '''
    return almostprime(n, 2)
    
def eulertotient(x: int | list) -> int:
    '''
    Evaluates Euler's Totient function.
    Input: `x: int` is prime factorised by Lucas A. Brown's primefac.py
           else `x: list` is assumed to the prime factorisation of `x: int`
    '''
    pfactors = x if isinstance(x, list) else factors(n)
    return prod((p-1)*(p**(e-1)) for (p, e) in pfactors)
# def eulertotient(n: int) -> int:
#     '''
#     Uses trial division to compute 
#     Euler's Totient (Phi) Function.
#     '''
#     phi = int(n > 1 and n)
#     for p in range(2, int(n ** .5) + 1):
#         if not n % p:
#             phi -= phi // p
#             while not n % p:
#                 n //= p
#     #if n is > 1 it means it is prime
#     if n > 1: phi -= phi // n 
#     return phi