def prob5():
factors = {}
for i in range(1,21):
prime_factors = utils.prime_factorization(i)
utils.dict_merge(factors, prime_factors)
product = utils.prime_defactorization(factors)
return product
def blum_blum_shub(p, q):
assert p % 4 == 3 and q % 4 == 3
n = p * q
possible_seeds = prime_factorization(n)
seed = random.choice(possible_seeds)
seeds = seed_generator(seed, n)
for s in seeds:
yield s % 2
def encryption_key(p, q):
#assert p % 3 == 4 and q % 3 == 4
phi_n = phi(p, q)
public_key = random.randint(1, 2 ** 128)
phi_factors = prime_factorization(phi_n)
while public_key in phi_factors:
public_key = random.randint(1, 2 ** 128)
return public_key
def prime_factorization(base, power):
factorization = utils.prime_factorization(base)
for factor in factorization:
factorization[factor] *= power
factorization_str = ''
for factor, power in factorization.items():
factorization_str += '{}^{},'.format(factor, power)
return factorization_str
def sum_proper_divisor(n):
pf = prime_factorization(n)
sum_d = 1
for p in pf:
d = 0
for i in range(pf[p] + 1):
d += p**i
sum_d *= d
sum_d -= n
return sum_d
def prob3():
prime_factors = utils.prime_factorization(600851475143)
return list(prime_factors)[0]
# Highly divisible triangular number
from utils import prime_factorization
max_divisor = 0
for i in range(2, 15000):
p1 = prime_factorization(i)
p2 = prime_factorization(i+1)
c = p1 + p2
divisor = 1
for count in c.values():
divisor *= (count+1)
if divisor > max_divisor:
max_divisor = divisor
print max_divisor
if divisor > 500:
print i, i*(i+1)/2, c
break
from collections import Counter
from functools import reduce
from operator import mul
import utils
high = 100000
utils.initialize_primes_cache(high)
# explanation: nth triangle number = n*(n-1)/2 = (n^2 - n)/2 < n^2
# highest prime factor of that can be sqrt
# so n is upper bound on highest prime factor of nth triangle number
s = 0
for i in range(1, high):
s += i
pf = Counter(utils.prime_factorization(s))
# explanation: we don't need to know the actual divisors, just how many of them there are
# itertools.combinations would let us enumerate the actual divisors
num_divisors = reduce(mul, (exp + 1 for (prime, exp) in pf.items()), 1)
print(s, num_divisors)
if num_divisors > 500: break
def random_near_primitive_root(p):
start = random.randint(2, p - 1)
primes_to_test = prime_factorization(p - 1)
for b in infinite_range(start, p):
if primitive_root(b, primes_to_test, p):
return b
def prob3():
prime_factors = utils.prime_factorization(600851475143)
return list(prime_factors)[0]
import sys
sys.path.append('../utils')
from utils import is_prime, prime_factorization
for x in range(10,200000):
a = prime_factorization(x)
if len(set(a)) == 4:
b = prime_factorization(x+1)
if len(set(b)) == 4:
c = prime_factorization(x+2)
if len(set(c)) == 4:
d = prime_factorization(x+3)
if len(set(d)) == 4:
print(str(x) + ': [%s]' % ', '.join(map(str, a)))
print(str(x+1) + ': [%s]' % ', '.join(map(str, b)))
print(str(x+2) + ': [%s]' % ', '.join(map(str, c)))
print(str(x+3) + ': [%s]' % ', '.join(map(str, d)))
assert 5 == 6
import utils
upper_bound = 200000
num_consec = 4
utils.initialize_primes_cache(upper_bound)
factors = dict((i, set(utils.prime_factorization(i))) for i in range(4, upper_bound))
for i in range(4, upper_bound):
if all(len(factors[i+j]) >= num_consec for j in range(num_consec)):
print(i, factors[i])
break