def special_num_factors(a, b, hash_):
factors = (prime_factors(a, unique=False, hash_=hash_) +
prime_factors(b, unique=False, hash_=hash_))
factors = list_frequencies(factors)
prod = 1
for factor in factors:
prod *= factor[1] + 1
return prod
def main(verbose=False):
product = 13082761331670030
factors = prime_factors(product)
candidate_lists = []
for factor in factors:
candidate_lists.append([(factor, root)
for root in find_cube_roots(factor)])
result = list(i_product(*candidate_lists))
coprime_units = {}
for factor in factors:
_, multiplier = extended_euclid(factor, product / factor)
coprime_units[factor] = multiplier * (product / factor)
vals = []
for pairing in result:
count = 0
for prime, residue in pairing:
count += residue * coprime_units[prime]
count = count % product
vals.append(count)
return sum(vals) - 1 # 1 is in there as (1,1,...,1)
def main(verbose=False):
prime_factors_hash = {}
MINIMUM_SOLUTIONS = 4*(10**6)
# P^k < 10**7 (10 mil)
powers = [power_up_to_digits(prime, 7)
for prime in [3, 5, 7]]
products = [reduce(operator.mul, triple) for
triple in list(i_product(*powers))]
products = [product for product in sorted(products)
if product > 2*MINIMUM_SOLUTIONS][:20]
PRIMES = sieve(100)
max_prod = 10**21
res = []
for product in products:
factors = prime_factors(product, unique=False, hash_=prime_factors_hash)
factors = [(factor - 1)/2 for factor in factors][::-1]
curr_prod = 1
for i, exp in enumerate(factors):
curr_prod = curr_prod*(PRIMES[i]**exp)
if curr_prod < max_prod:
max_prod = curr_prod
return max_prod
def main(verbose=False):
# Find the first four consecutive integers to have four distinct
# primes factors. What is the first of these numbers?
factor_hash = {1: [], 2: [2]}
# Smallest product of 4 primes is 2*3*5*7 = 210
# We need to update the hash to get to this point
for i in range(3,210 + 1):
prime_factors(i, hash_=factor_hash)
smallest = 210 # The smallest integer of the four
num_factors = [ len(prime_factors(smallest + i,
unique=True,
hash_=factor_hash))
for i in range(4) ]
while num_factors != [4,4,4,4]:
smallest = increment(smallest, num_factors)
num_factors = [ len(prime_factors(smallest + i,
unique=True,
hash_=factor_hash))
for i in range(4) ]
return smallest
def main(verbose=False):
return max(prime_factors(600851475143))
