from bisect import bisect_left
def genDoubleSquares(bound):
for i in range(1, bound):
yield i ** 2 * 2
def oddCompositeSieve(bound):
sieve = [True] * bound
sieve[0] = sieve[1] = False
for (i, prime) in enumerate(sieve):
if prime:
for n in range(i * i, bound, i):
sieve[n] = False
elif i % 2 == 1 and i > 1:
yield i
bound = 10000
primes = list(primeSieve(bound))
doubleSquares = list(genDoubleSquares(bound))
for i in oddCompositeSieve(bound):
summable = False
for j in primes[:bisect_left(primes, i)]:
for k in doubleSquares[:bisect_left(doubleSquares, i)]:
if j + k == i:
summable = True
break
if summable:
break
if not summable:
print(i)
break
#=============================================================================== # Find the sum of all the primes below two million. #=============================================================================== from Common import primeSieve print(sum(primeSieve(2000000)))
#
# d2d3d4=406 is divisible by 2
# d3d4d5=063 is divisible by 3
# d4d5d6=635 is divisible by 5
# d5d6d7=357 is divisible by 7
# d6d7d8=572 is divisible by 11
# d7d8d9=728 is divisible by 13
# d8d9d10=289 is divisible by 17
#
# Find the sum of all 0 to 9 pandigital numbers with this property.
#===============================================================================
from Common import primeSieve
from itertools import permutations
primes = [0] + list(primeSieve(18))
pandigitals = list(permutations(range(10)))
def isSpecial(n):
s = str(n)
for i in range(1, 8):
sub = s[i] + s[i + 1] + s[i + 2]
if not int(sub) % primes[i] == 0:
return False
return True
total = 0
for t in pandigitals:
t = str(t).replace(', ', '').replace('(', '').replace(')', '')
if isSpecial(t):
total += int(t)
#===============================================================================
# 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 Common import primeSieve
cache = {}
for i in primeSieve(10000):
s = str(i)
if len(s) == 4:
ordered = ''.join(sorted(s))
if ordered in cache:
cache[ordered].append(i)
else:
cache[ordered] = [i]
for k in cache:
perms = cache[k]
if len(perms) >= 3:
diffs = {}
for i in range(len(perms)):
for j in range(i + 1, len(perms)):
diff = perms[j] - perms[i]
if diff in diffs:
if not str(perms[i]) in diffs[diff]:
#===============================================================================
# Find the sum of the only eleven primes that are both truncatable from left to right and right to left.
#===============================================================================
from Common import isPrime, primeSieve
def isTruncatablePrime(n):
if n <= 7:
return False
s = str(n)
length = len(s)
for i in range(length):
if not isPrime(int(s[i:])):
return False
for i in range(length):
if not isPrime(int(s[:length - i])):
return False
return True
print(sum([i for i in primeSieve(1000000) if isTruncatablePrime(i)]))
#===============================================================================
# How many circular primes are there below one million?
#===============================================================================
from Common import isPrime, primeSieve
def isCircularPrime(n):
s = str(n)
for i in range(len(s)):
if not isPrime(int(s[i:] + s[:i])):
return False
return True
print(len([i for i in primeSieve(1000000) if isCircularPrime(i)]))
# =============================================================================== # What is the largest n-digit pandigital prime that exists? # =============================================================================== from Common import primeSieve, isPandigital print(max([i for i in primeSieve(10000000) if isPandigital(i)]))
#
# d2d3d4=406 is divisible by 2
# d3d4d5=063 is divisible by 3
# d4d5d6=635 is divisible by 5
# d5d6d7=357 is divisible by 7
# d6d7d8=572 is divisible by 11
# d7d8d9=728 is divisible by 13
# d8d9d10=289 is divisible by 17
#
# Find the sum of all 0 to 9 pandigital numbers with this property.
#===============================================================================
from Common import primeSieve
from itertools import permutations
primes = [0] + list(primeSieve(18))
pandigitals = list(permutations(range(10)))
def isSpecial(n):
s = str(n)
for i in range(1, 8):
sub = s[i] + s[i + 1] + s[i + 2]
if not int(sub) % primes[i] == 0:
return False
return True
total = 0
for t in pandigitals:
t = str(t).replace(', ', '').replace('(', '').replace(')', '')
