def prime_numbers(**kwargs):
from maths.misc import is_prime
limit = kwargs.get('limit', None)
start = kwargs.get('start', 1)
if limit is None:
import itertools
for i in itertools.count(start,1):
if is_prime(i):
yield i
else:
found_primes = 0
i = start
while found_primes < limit:
if is_prime(i):
found_primes += 1
yield i
i += 1
num_digits = len(digits)
for i in xrange(1, num_digits):
# Remove digits from the right
yield digit_list_to_num(digits[:i])
# Remove digits from the left
yield digit_list_to_num(digits[i:])
if __name__ == "__main__":
matching_primes = []
from maths.misc import is_prime
from maths.sequences import prime_numbers
exclusion_set = (2, 3, 5, 7)
# Start prime search at 10
for prime_number in prime_numbers(limit = None, start = 10):
if len(matching_primes) > 10:
break
found_prime = True
for n in remove_digits(prime_number):
if n is 1 or not is_prime(n):
found_prime = False
break
if found_prime:
matching_primes.append(prime_number)
print('Matching primes: %s'%matching_primes)
print('The sum of matching primes is %d'%sum(matching_primes))
def list_to_num(digits):
length = len(digits)
num = 0
for i in xrange(length):
power = length - 1 - i
num += digits[i] * 10**power
return num
if __name__ == "__main__":
from maths.misc import get_digits, is_prime
circular_primes = []
for i in xrange(2, 1000000):
if not is_prime(i):
continue
prime = True
digits = get_digits(i)
for offset in xrange(1,len(digits)):
shifted = shift_digits(digits, offset)
num = list_to_num(shifted)
if not is_prime(num):
prime = False
break
if prime:
circular_primes.append(i)
print('The number of circular primes below 1,000,000 is %d'%len(circular_primes))
Considering quadratics of the form:
n^2 + an + b, where |a| < 1000 and |b| < 1000
where |n| is the modulus/absolute value of n
e.g. |11| = 11 and |-4| = 4
Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0.
"""
if __name__ == "__main__":
best_a, best_b, best_n = None, None, 0
largest_prime = -1
for a in xrange(-1000, 1001, 1):
for b in xrange(-1000, 1001, 1):
from maths.misc import is_prime
prime = True
n = 0
while prime:
result = n*n + a*n + b
prime = is_prime(abs(result))
if not prime:
break
n += 1
if n > best_n:
best_a, best_b, best_n = a, b, n
print('(a,b) = (%d, %d) with n = %d with a*b=%d'%(best_a, best_b, best_n, best_a*best_b))
