from eratosthenes import sieve
# How far to go?
primes = sieve(1000000)
prime_set = set(primes)
print 'generated primes'
num_primes = 0
s = 0
for p in primes:
if p <= 7:
continue
# truncate right
passed = True
p_str = list(str(p))
while len(p_str) >= 2:
p_str.pop()
if int(''.join(p_str)) not in prime_set:
passed = False
break
if not passed:
continue
# truncate left
passed = True
p_str = list(str(p))
while len(p_str) >= 2:
p_str.pop(0)
if int(''.join(p_str)) not in prime_set:
passed = False
# Find numbers a, b of the form n^2 + an + b, where the most primes are
# produced with consecutive values of n starting with 0,
# for a, b ranging between -1000 and 1000
from eratosthenes import sieve
# go up to 1000^2 + 1000*1000 + 1000 == 2001000
limit = 2001000
primes = set(sieve(limit))
max_n = 0
the_a = None
the_b = None
for a in range(-1000, 1001):
for b in range(-1000, 1001):
n = 0
eq = n*n + a*n + b
while eq in primes:
n += 1
eq = n*n + a*n + b
if n > max_n:
max_n = n
the_a = a
the_b = b
print max_n, the_a, the_b
print the_a * the_b
generated by a single quadratic q.
These bounds on a, b, and n imply q(n) < 3e6. Say 4e6, just to be safe.
The fastest way to check primality of numbers in the 10^6 range is to enumerate
them all with a sieve.
'''
if path.exists(__cache__):
primes = np.load(__cache__)
is_prime = np.zeros((5e6), dtype='bool')
is_prime[primes] = True
else:
is_prime = np.ones((5e6), dtype='int8')
eratosthenes.sieve(is_prime)
is_prime.dtype = 'bool'
primes = np.where(is_prime)[0]
np.save(__cache__, primes)
def candidates(max_a, max_b):
'''
candidates(int max_a=1000, int max_b=1000) yields (int, int)
Given the bounds |a| < max_a and |b| < max_b, yields candidates for
(a, b) as described above.
'''
for b in takewhile(lambda x: x < max_b, primes):
for q in takewhile(lambda x: x < 1 + max_a + b, primes):
yield (q - 1 - b, b)
from eratosthenes import sieve
s = sieve(1000000)
prime_set = set(s)
summands = sieve(50000)
consecutives = 0
the_prime = 0
print 'evaluating', len(summands), 'summands'
for i, p in enumerate(summands):
p_list = []
for j in range(i, len(summands)):
p_list.append(summands[j])
the_sum = sum(p_list)
if the_sum in prime_set:
if len(p_list) > consecutives:
consecutives = len(p_list)
the_prime = the_sum
print the_prime, consecutives
from eratosthenes import sieve
import itertools
s = sieve(1000000)
prime_set = set(s)
def replace(s, i, n):
"""
Replace the character at index i in the string s with the value n
>>> replace('12345', 0, '8')
'82345'
>>> replace('12345', 1, '8')
'18345'
>>> replace('12345', 2, '8')
'12845'
>>> replace('12345', 3, '8')
'12385'
>>> replace('12345', 4, '8')
'12348'
"""
return s[:i] + n + s[i+1:]
def num_primes_in_family(x):
"""
from eratosthenes import sieve
import math
limit = 10e6
primes = set(sieve(int(limit)))
print 'genned primes'
def decompose(n):
"""
Decompose the composite number n into the sum of a prime and twice a
square. Return whether it was successful.
2 x i^2 + p = n
2 x i^2 = n - p
2 x i^2 <= n - 2
i^2 <= ((n-2)/2)
i < sqrt((n-2)/2)
"""
l = int(math.ceil(math.sqrt((n-2)/2.)))
for p_sq in range(1, l):
if n - (2 * p_sq * p_sq) in primes:
return True
return False
for n in range(37, int(limit), 2):
if n % 2 == 0:
continue
if n in primes:
continue
from eratosthenes import sieve
s = set(sieve(10000))
inc = 3330
for i in range(999, 10000, 2):
i2 = i + 3330
i3 = i + 6660
if i in s and i2 in s and i3 in s:
si1 = set(list(str(i)))
si2 = set(list(str(i2)))
si3 = set(list(str(i3)))
if si1 == si2 and si2 == si3:
print '{0}{1}{2}'.format(i, i+3330, i+6660)
import numpy as np import pyximport pyximport.install() import eratosthenes A = np.ones(5000000, dtype='int8') eratosthenes.sieve(A) A.dtype = 'bool' assert not A[41 * 41] print np.where(A)[0]
