#!/usr/bin/env python
"""
Problem 007:
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that
the 6th prime is 13.
What is the 10001st prime number?
"""
from euler import postponed_sieve
N = 10001
p = postponed_sieve()
for i in range(N - 1):
next(p)
print(next(p))
#!/usr/bin/env python
"""
Problem 007:
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that
the 6th prime is 13.
What is the 10001st prime number?
"""
from euler import postponed_sieve
N = 10001
p = postponed_sieve()
for i in range(N - 1):
next(p)
print(next(p))
#!/usr/bin/env python
"""
Problem 123:
Let p_n be the nth prime: 2, 3, 5, 7, 11, ..., and let r be the remainder when
(p_n - 1)^n + (p_n + 1)^n is divided by p_n^2.
For example, when n = 3, p_3 = 5, and 4^3 + 6^3 = 280 - 5 mod 25.
The least value of n for which the remainder first exceeds 10^9 is 7037.
Find the least value of n for which the remainder first exceeds 10^10.
"""
from euler import postponed_sieve
N = 10**10
primes = postponed_sieve()
n, p = 1, next(primes)
while (pow(p - 1, n, p * p) + pow(p + 1, n, p * p)) % (p * p) < N:
n += 1
p = next(primes)
print(n)
