'''
The trial division algorithm is essentially the idea that
all factors of an integer n are less than or equal to isqrt(n),
where isqrt is floor(sqrt(n)).

Moreover, if p divides n, then all other factors of n must
be factors of n//p. Hence they must be <= isqrt(n//p).
'''
from math import isqrt # integer square root

# Returns the "multiplicity" of a prime factor
def pf_multiplicity(n: int, p: int) -> int:
    mult = 0
    while n % p == 0: 
        n //= p
        mult += 1
    return n, mult

'''
Trial division prime factorisation algorithm.
Returns a list of tuples (p, m) where p is
a prime factor and m is its multiplicity.
'''
def trial_division(n: int) -> list[tuple[int, int]]:
    factors = []

    # determine multiplicity of the only even prime (2)
    n, mult_2 = pf_multiplicity(n, 2)
    if mult_2: factors.append((2, mult_2))
    # determine odd factors and their multiplicities
    p = 3
    mult = 0
    limit = isqrt(n)
    while p <= limit:
        n, mult = pf_multiplicity(n, p)
        if mult:
            factors.append((p, mult))
            limit = isqrt(n) # recalculate limit
            mult = 0 # reset
        else:
            p += 2
    # if n is still greater than 1, then n is a prime factor
    if n > 1:
        factors.append((n, 1))
    return factors