"""
Project Euler Problem 41
========================
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?
"""
import euler_utils as utils
primes = utils.primes_till((10**7))
ndigit = [str(i) for i in range(1,10)]
#print ndigit
for i in range(len(primes)-1, 1, -1):
number_str = str(primes[i])
length = len(number_str)
if len(set(number_str).intersection(set(ndigit[0:length]))) == length:
print primes[i]
break
"""
Project Euler Problem 35
========================
The number, 197, is called a circular prime because all rotations of the
digits: 197, 971, and 719, are themselves prime.
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?
"""
import euler_utils as utils
primes = set(utils.primes_till(1000000))
count = 0
for num in primes:
num_str = str(num)
if all([x in primes for x in [int(num_str[i:]+num_str[:i])
for i in range(len(num_str))]]):
count += 1
print count
Let d(n) be defined as the sum of proper divisors of n (numbers less than
n which divide evenly into n).
If d(a) = b and d(b) = a, where a =/= b, then a and b are an amicable pair
and each of a and b are called amicable numbers.
For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22,
44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1,
2, 4, 71 and 142; so d(284) = 220.
Evaluate the sum of all the amicable numbers under 10000.
"""
import euler_utils as utils
no_op_set = set(utils.primes_till(10000))
dn_map = {}
amicable_pairs = []
dn = lambda x:sum(utils.divisors_of(x))
for num in range(4,10001):
if num not in no_op_set and num not in dn_map:
dsum = dn(num)
dn_map[num] = dsum
if dsum < 10000 and dsum > num:
dsum_sum = dn(dsum)
if dsum_sum == num:
amicable_pairs.append((num, dsum))
dn_map[dsum] = dsum_sum