def main():
size = 1000000
sieve = get_sieve(size)
seen = set([])
required_family_size = 8
smallest = size
for num in xrange(size):
if sieve[num] == 0 or num in seen:
continue
str_num = str(num)
for i in xrange(10):
char = str(i)
if sieve[num] == 1:
family = [num]
seen.add(num)
else:
family = []
if char in str_num:
for j in xrange(10):
if j != i:
new_num = int(str_num.replace(char, str(j)))
if len(str(new_num)) != len(str_num):
continue
if sieve[new_num] == 1:
family.append(new_num)
seen.add(new_num)
if len(family) >= required_family_size:
family.sort()
if family[0] < smallest:
smallest = family[0]
print smallest
25 = 7 + 2 * 3^2
27 = 19 + 2 * 2^2
33 = 31 + 2 * 1^2
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a
prime and twice a square?
"""
import math
from problem47 import get_sieve
max_num = 100000
sieve = get_sieve(max_num)
def get_smaller_prime(num):
for i in xrange(num - 1, 0, -1):
if sieve[i] == 1:
return i
# throw an error
return 0
def is_goldbach(num):
prime = get_smaller_prime(num)
while prime > 1:
new_num = num - prime
if new_num % 2 == 0:
sqrt = math.sqrt(new_num / 2)
if sqrt == int(sqrt):
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right
diagonal, but what is more interesting is that 8 out of the 13 numbers lying
along both diagonals are prime; that is, a ratio of 8/13 62%.
If one complete new layer is wrapped around the spiral above, a square spiral
with side length 9 will be formed. If this process is continued, what is the
side length of the square spiral for which the ratio of primes along both
diagonals first falls below 10%?
"""
from math import sqrt
from problem47 import get_sieve
sieve = get_sieve(10000000)
def is_prime(num):
if num < len(sieve):
return sieve[num] == 1
limit = int(sqrt(num)) + 1
if limit > len(sieve):
print "Error: sieve not large enough"
return False
for i in xrange(2, limit):
if sieve[i] == 1 and num % i == 0:
return False
return True
25 = 7 + 2 * 3^2
27 = 19 + 2 * 2^2
33 = 31 + 2 * 1^2
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a
prime and twice a square?
"""
import math
from problem47 import get_sieve
max_num = 100000
sieve = get_sieve(max_num)
def get_smaller_prime(num):
for i in xrange(num - 1, 0, -1):
if sieve[i] == 1:
return i
# throw an error
return 0
def is_goldbach(num):
prime = get_smaller_prime(num)
while prime > 1:
new_num = num - prime
if new_num % 2 == 0:
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?
for each i in the set
new_set = permuations(rest of set - i)
return_set = set()
for item in set:
return_set.add(i + item)
return set
"""
from problem47 import get_sieve
sieve = get_sieve(10000)
def get_permutations(num):
def _get_permutations_helper(str_num):
if not str_num:
return set([""])
permutations = set()
for i in xrange(len(str_num)):
new_num = list(str_num)
head = new_num.pop(i)
temp_permutations = _get_permutations_helper(new_num)
for permutation in temp_permutations:
permutations.add(head + permutation)
return permutations
