def generate_primes():
n = 1
while True:
while not t.is_prime(n):
n += 1
yield n
n += 1
def left_extend(prefix, curr):
num = int(prefix + curr)
if not t.is_prime(num):
return
if is_truncatable_prime(str(num)):
truncatable_primes.append(num)
for digit in extender_digits:
left_extend(str(digit), prefix + curr)
def euler_totient(n):
totient = n
if t.is_prime(n):
totient = n - 1
else:
for prime in prime_divisors[n]:
totient = totient // prime
totient *= prime - 1
euler_totients.append((n, totient))
return (n, totient)
def add_all_pandigital_prime_sets(permutation):
perm_partitions = get_all_partitions(permutation, 0)
for partition in perm_partitions:
candidate = frozenset(sorted(partition))
if candidate in unique_non_prime or candidate in unique:
continue
if all(t.is_prime(int(x)) for x in partition):
unique.add(candidate)
else:
unique_non_prime.add(candidate)
def is_prime(x):
if x in primes:
return True
elif x in nonprimes:
return False
elif t.is_prime(x):
primes.append(x)
return True
else:
nonprimes.append(x)
return False
def get_prime_factorization_of_(n):
prime_factors = []
if t.is_prime(n):
primes.append(n)
prime_factors.append(n)
return prime_factors
n_copy = n
for prime in primes:
while n_copy % prime == 0:
prime_factors.append(prime)
n_copy //= prime
if n_copy == 1:
return prime_factors
def is_prime(x):
if x in primes:
return True
elif t.is_prime(x):
return True
return False
#!/usr/bin/env python3 -tt
import naive_prime_test as t
import numpy as np
primes = [x for x in range(1, 1002) if t.is_prime(x)]
def prime_divisors(n):
return [prime for prime in primes if n % prime == 0]
def euler_totient(n):
totient = n
for prime in prime_divisors(n):
totient = totient // prime
totient *= prime - 1
return totient
print(
np.argmax(np.array([n / euler_totient(n) for n in range(1, 1000000)])) + 1)
def is_right_truncatably_prime(x):
return all(t.is_prime(int(x[i:])) for i in range(len(x)))
def is_left_truncatably_prime(x):
return all(t.is_prime(int(x[:len(x) - i])) for i in range(len(x)))
n_copy = n
for prime in primes:
while n_copy % prime == 0:
prime_factors.append(prime)
n_copy //= prime
if n_copy == 1:
return prime_factors
all_prime_factors = [
get_prime_factorization_of_(n) for n in range(2, upper_bound)
]
for i in range(upper_bound - 2):
n = i + 2
if t.is_prime(n):
factorization_book[n] = {(n, )}
continue
factorizations = set()
for pf in set(all_prime_factors[i]):
factors_without_pf = n // pf
factorizations_wo_pf = factorization_book[factors_without_pf]
for factorization_wo_pf in factorizations_wo_pf:
lst = sorted([pf] + list(factorization_wo_pf))
factorizations.add(tuple(lst))
fact_wo_pf_as_list = list(factorization_wo_pf)
for index in range(len(fact_wo_pf_as_list)):
lst = fact_wo_pf_as_list[:]
lst[index] *= pf
factorizations.add(tuple(sorted(lst)))
factorization_book[n] = factorizations
#!/usr/bin/env python3 -tt
import naive_prime_test as t
import numpy as np
prime_divisors = {}
def list_digits(x):
return "".join(sorted(str(x)))
def have_same_digits(x, y):
return list_digits(x) == list_digits(y) and len(str(x)) == len(str(y))
primes = [x for x in range(1,10000000//3 + 1) if t.is_prime(x)]
print("got primes")
def get_prime_divisors(n):
results = []
n_copy = n
for prime in primes:
if n % prime == 0:
results.append(prime)
n_copy = n_copy // prime
results.extend(prime_divisors[n_copy])
prime_divisors[n] = results
prime_divisors[1] = []
for i in range(2, 10000000):
get_prime_divisors(i)
#!/usr/bin/env python3 -tt import math import naive_prime_test as t # Find the sum of all the primes below two million. print(sum([x for x in range(2000000) if t.is_prime(x)]))
#!/usr/bin/env python3 -tt
import math
import naive_prime_test as t
first_five = [1, 3, 5, 7, 9]
last_prime = first_five[-1]
total_primes = len(list(filter(t.is_prime, first_five)))
total_diags = len([1, 3, 5, 7, 9])
i = 2
while total_primes / total_diags > 0.10:
i += 2
for _ in range(4):
last_prime += i
if t.is_prime(last_prime):
total_primes += 1
total_diags += 4
print(i + 1)