def is_trunctable_prime(n):
if not is_prime(n):
return False
n = str(n)
for i in range(1, len(n)):
if not is_prime(int(n[i:])) or not is_prime(int(n[:-i])):
return False
return True
def prime_factors(n):
if is_prime(n):
return [n]
factors = []
for num in range(2, n // 2 + 1):
if is_prime(num):
if n % num == 0:
factors.append(num)
while n % num == 0:
n //= num
if n == 1:
break
continue
return factors
#
# There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.
#
# How many circular primes are there below one million?
from Problem_26 import is_prime
def rotations(n):
digits = [int(d) for d in str(n)]
permutations = []
while digits not in permutations:
permutations.append(digits.copy())
digits.insert(0, digits.pop())
return permutations
if __name__ == '__main__':
circular_primes = []
for n in range(2, 1000000):
if is_prime(n):
rots = [is_prime(int("".join([str(d) for d in rotation]))) for rotation in rotations(n)]
if False not in rots:
temp_list = []
for rotation in rotations(n):
temp_list.append(int("".join([str(d) for d in rotation])))
print(n, temp_list, rots)
circular_primes.append(n)
print(circular_primes)
print("There are *{}* circular primes below one million.".format(len(circular_primes)))
# Pandigital prime
#
# We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once.
# For example, 2143 is a 4-digit pandigital and is also prime.
#
# What is the largest n-digit pandigital prime that exists?
from Problem_26 import is_prime
from Problem_32 import is_pandigital
largest_n_digit_pandigital_prime = 1
# start from 7654321 because {987654321} (9-digit pandigitals) and {87654321} are divisible by 3 and so not prime
# remember that if the sum of a number's digits is divisible by 3, the number is also divisible by 3
n = 7654321
while largest_n_digit_pandigital_prime == 1:
if is_prime(n) and is_pandigital(n, n_max=len(str(n))):
print(n, "is prime and is a {}-digit pandigital".format(len(str(n))))
largest_n_digit_pandigital_prime = n
break
n -= 2
print(largest_n_digit_pandigital_prime)
from Problem_26 import is_prime
max_primes_generated = 0
max_a = 0
max_b = 0
for a in range(-999, 999, 2):
for b in range(-1000, 1000):
if not is_prime(abs(b)):
continue
primes_generated = 0
n = 0
while is_prime(n**2 + a * n + b):
primes_generated += 1
n += 1
if primes_generated > max_primes_generated:
max_primes_generated = primes_generated
max_a = a
max_b = b
if primes_generated > 10:
print("n**2 + " + str(a) + "n + " + str(b))
print(primes_generated)
print(max_primes_generated, max_a, max_b, max_a * max_b)
# Prime permutations
#
# The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways:
# (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.
#
# There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property,
# but there is one other 4-digit increasing sequence.
#
# What 12-digit number do you form by concatenating the three terms in this sequence?
from Problem_26 import is_prime
primes = {}
for n in range(1234, 9877):
if is_prime(n):
digits = ''.join(sorted(str(n)))
if digits in primes.keys():
primes[digits].append(n)
else:
primes[digits] = [n]
print(len(primes))
primes = dict(filter(lambda x: len(x[1]) > 2, primes.items()))
print(len(primes), primes)
for permutations in primes.values():
for p1 in range(len(permutations) - 2):
if permutations[p1] + 3330 in permutations and permutations[
p1] + 6660 in permutations:
print(str(permutations[p1]), str(permutations[p1] + 3330),
str(permutations[p1] + 6660))