def truncatable(n):
if not is_prime(n):
return False
for x in xrange(1,len(str(n))):
if not is_prime(int(str(n)[:x])) or not is_prime(int(str(n)[x:])):
return False
return True
def num_of_divisor(n):
d = dict()
cur = 2
while True:
if util.is_prime(cur):
d[cur] = 0
while n % cur == 0:
n /= cur
d[cur] += 1
n = int(n)
if n == 1:
break
elif util.is_prime(n):
d[n] = 1
break
if cur == 2:
cur += 1
else:
cur += 2
ans = 1
for (_, v) in d.items():
ans *= v + 1
return ans
def solve():
for p in primes2(3340):
if p < 1000:
continue
p1 = p + 3330
p2 = p + 6660
if not is_prime(p1) or not is_prime(p2):
continue
if sorted(str(p)) == sorted(str(p1)) == sorted(str(p2)):
return str(p) + str(p1) + str(p2)
def is_prime_pair(a, b):
ab = int(str(a) + str(b))
if not util.is_prime(ab):
return False
ba = int(str(b) + str(a))
if not util.is_prime(ba):
return False
return True
def is_removeable_prime(n):
if not is_prime(n):
return False
sn = str(n)
for i in range(1, len(sn)):
if not is_prime(int(sn[i:])):
return False
if not is_prime(int(sn[:i])):
return False
return True
def is_truncatable_prime(n):
if str(n)[0] not in ['2', '3', '5', '7'
] or str(n)[-1] not in ['2', '3', '5', '7']:
return False
if not is_prime(n):
return False
n = str(n)
truncs = [int(n[i:]) for i in range(len(n))
] + [int(n[:i + 1]) for i in range(len(n))]
return all([is_prime(k) for k in truncs])
def findPairs(i, primes):
'''
@param i Index of primes
'''
pairs = set()
# startTime = time.time()
for j in range(i, len(primes)):
if is_prime(concatenate(primes[i], primes[j])) and is_prime(concatenate(primes[j], primes[i])):
pairs.add(primes[j])
# print('calculate pair[{0}] takes {1} seconds'.format(primes[i], time.time()-startTime))
return pairs
def solve():
p = 33
while 1:
p += 2
if not is_prime(p):
for i in range(1, p / 2 + 1):
if is_prime(p - 2 * i * i):
break
else:
return p
def is_circular_prime(n):
if not is_prime(n):
return False
digits = list(str(n))
checklist = ['0', '4', '6', '8']
if any([d in checklist for d in digits]):
return False
cycles = generate_rotation(n)
while True:
try:
if not is_prime(int(next(cycles))):
return False
except StopIteration:
return True
def are_all_rotations_prime(n):
if n < 2:
return False
if not util.is_prime(n):
return False
s = str(n)
for i in range(len(s)):
s2 = s[i:] + s[:i]
if not util.is_prime(int(s2)):
return False
print '!!', n
return True
def get_prime_factors(n):
result = set()
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
if is_prime(i):
result.add(i)
upper_divisor = int(n / i)
if is_prime(upper_divisor):
result.add(upper_divisor)
if is_prime(n):
result.add(n)
return list(result)
def is_truncatable_prime(n):
num, ds = digits(n)
# remove digits from left to right
for i in xrange(1, len(ds)):
tmp_ds = ds[i:]
tmp_n = int(reduce(lambda x, y: ''.join([str(x), str(y)]), tmp_ds, ''))
if not is_prime(tmp_n):
return False
# remove digits from right to left
for i in xrange(1, len(ds)):
tmp_ds = ds[:len(ds)-i]
tmp_n = int(reduce(lambda x, y: ''.join([str(x), str(y)]), tmp_ds, ''))
if not is_prime(tmp_n):
return False
return True
def count_prime(a, b):
n = 0
while 1:
if is_prime(abs(n * n + a * n + b)):
n += 1
else:
return n
def main():
ans = 0
for i in permutations("123456"):
temp = int(num+"".join(i))
if is_prime(temp) and temp > ans :
ans = temp
print ans
def solution():
count = 1
n = 3
while count < 10001:
if is_prime(n):
count += 1
n += 2
return n - 2
def find_primes_ceiling(self, starting_prime, ceiling) -> list:
out_list = []
for current_candidate in range(starting_prime, ceiling, 2):
if util.is_prime(current_candidate):
out_list.append(current_candidate)
print("search ceiling exceeded, primes found this run {0:,.0f}".format(
len(out_list)))
return out_list
def solution():
count = 1
n = 3
while count < 10001:
if is_prime(n):
count += 1
n += 2
return n - 2
def consecutive_primes(a, b):
"""
Find the number of consecutive primes generated by the formula
n² + an + b
"""
n = 0
while True:
if not is_prime(n * n + a * n + b):
return n
n += 1
def get_deepest_prime_below(sums, below):
rr = []
for y in reversed(xrange(len(sums))):
for x in range(len(sums[y])):
n = sums[y][x]
if util.is_prime(n) and n < below:
rr.append(n)
if rr:
break
return rr
def nth_prime(n):
prime = 2
count = 1
k = 3
while count < n:
if is_prime(k):
prime = k
count += 1
k += 2
return prime
def consecutive_primes(a, b):
"""
Find the number of consecutive primes generated by the formula
n² + an + b
"""
n = 0
while True:
if not is_prime(n*n + a*n + b):
return n
n += 1
def next(x):
new_xs = []
str_x = str(x)
for (i,digit) in enumerate(str_x + '0'):
for d in range(0,10):
temp = str_x[:i] + str(d) + str_x[i:]
if not (i == 0 and d == 0):
if util.is_prime(int(temp)):
new_xs.append(int(temp))
return new_xs
def solution():
target = 2e6
i = 3
total = 2
while True:
if is_prime(i):
total += i
i += 2
if i > target:
break
return total
def p131():
# n**3 + p = (n+1)**3
# p = 3n**2 + 3n + 1
cnt = 0
for n in count(1):
p = 3 * (n**2) + 3 * n + 1
if p >= 1000000:
break
if is_prime(p):
cnt += 1
print(cnt)
def solution():
count = 1
i = 3
target = 10001
while(True):
if is_prime(i):
count += 1
if count == target:
break
i += 2
return i
def pandigital_prime(n_digits):
if n_digits % 2 == 0:
start = int(str(n_digits) * n_digits) + 1
else:
start = int(str(n_digits) * n_digits)
end = 10**(n_digits - 1)
digits = set("".join(str(d) for d in xrange(1, n_digits + 1)))
for i in xrange(start, end, -2):
if set(str(i)) == digits:
if util.is_prime(i):
return i
def main():
i = 0
n = 1
while i < 10001:
n += 1
if is_prime(n):
i += 1
print(n)
def test_keygen(self):
pk, sk = self.generate_keypair(_N_BITS)
# check p and q are actually safe primes
self.assertGreater(sk.p, 0)
self.assertGreater(sk.q, 0)
self.assertTrue(util.is_prime(sk.p))
self.assertTrue(util.is_prime(sk.q))
self.assertTrue(util.is_prime((sk.p - 1) // 2))
self.assertTrue(util.is_prime((sk.q - 1) // 2))
# check their sizes
self.assertEqual(sk.p.bit_length() + sk.q.bit_length(), _N_BITS)
self.assertGreaterEqual(sk.p.bit_length(), _N_BITS // 2)
self.assertGreaterEqual(sk.q.bit_length(), _N_BITS // 2)
# check consistency of n, nsquare and g
self.assertEqual(pk.n, sk.p * sk.q)
self.assertEqual(pk.nsquare, pk.n**2)
self.assertGreater(pk.g, 0)
self.assertLess(pk.g, pk.nsquare)
def is_truncatable_prime(string, left):
n = int(string)
if is_prime(n):
if n < 10:
return True
else:
next = string[0:-1]
if left:
next = string[1::]
return is_truncatable_prime(next, left)
else:
return False
def is_circular(n):
if n in ps:
return True
s = str(n)
ls = []
for i in range(len(s)):
if not is_prime(int(s)):
return False
ls.append(int(s))
s = s[1:] + s[0]
ps.extend(ls)
return True
def solve():
"""
Generate permutations of range(1, n) and concatenate to form pandigital
numbers. Start with n = 10 and count down until a prime pandigital is
found.
"""
for i in range(7, 1, -1):
for p in itertools.permutations(range(i, 0, -1)):
num = int("".join(str(i) for i in p))
if (num % 2 != 0) and util.is_prime(num):
print num
return
def main():
prime_perc = 100
total = 1
num_prime = 0
i = 1
while prime_perc >= .10:
i+=2
total += 4
for d in diagonal(i):
num_prime += is_prime(d)
prime_perc = float(num_prime)/total
print i,prime_perc
def get_xth_prime(xth):
primes_found = 0
x = 2
while True:
if is_prime(x):
primes_found += 1
if primes_found == xth:
return x
x += 1
def prime_proof(str_n):
l = len(str_n)
for i in range(l):
ln = list(map(int, str_n))
ln[i] = 0
c = int(reduce(lambda x, y: str(x) + str(y), ln))
# print(c, l - i - 1)
for j in range(10):
r = c + j * 10**(l - i - 1)
if is_prime(r):
return False
# print('\t', r)
return True
def circular_primes(max_n):
res = []
for i in itertools.chain((2,), xrange(3, max_n + 1, 2)):
s = str(i)
l = len(s)
if l > 1 and any([EVENS.search(s), "5" in s]):
continue
permutations = [int("".join(p)) for p in itertools.permutations(s, l)]
for p in permutations:
if not util.is_prime(p):
break
else:
res.extend(permutations)
return res
def ppd():
"pandigital primes descending"
from itertools import permutations
from math import floor
from util import is_prime
for n in range(0, 9):
digits = '987654321'[n:]
if sum(map(int, digits)) % 3 == 0:
# every permutation will be divisible by 3
continue
for p in permutations(digits):
x = int(''.join(p))
if is_prime(x):
yield x
def __init__(self, n: IntType) -> None:
super().__init__()
self.n = int(n)
assert n >= 1
self._props[RingProperties.FINITE] = True
self._props[RingProperties.FIELD] = is_prime(n)
self._props[RingProperties.ORDERED] = False
# Register coercions. Only one we've got is from ZZ.
def ZZ_coercion(x: IntegerElt) -> ZModElt:
return self(x.value)
self._register_coercion(ZZ, ZZ_coercion)
def p179(): # 60 min > run time (bad..)
L = 10**7
cnt = 1 # 2, 3
last_fc = 0
for i in range(2, L):
if is_prime(i):
last_fc = 0
continue
fc = factor_cnt(i)
if last_fc == fc:
cnt += 1
print([i], fc, last_fc)
last_fc = fc
print('[179]: ', cnt)
return
def main(ceiling):
"""Tests to see if next number in sequence 6i+-1 is prime"""
primes = [2, 3]
i = 1
switch = True
while primes[-1] <= ceiling:
if switch:
n = 6*i - 1
switch = False
else:
n = 6*i + 1
switch = True
i+=1
if is_prime(n, primes):
primes.append(n)
#print primes[-1]
return primes
def find_primes_timelimit(self, starting_prime, timelimit_seconds) -> list:
start_time = dt.datetime.now()
print(
f"find primes timelimit start time: {start_time.strftime('%H:%m')}"
)
current_candidate = starting_prime + 2
out_list = []
while (dt.datetime.now() - start_time).seconds < timelimit_seconds:
if util.is_prime(current_candidate):
out_list.append(current_candidate)
current_candidate += 2
print("search time limit exceeded, primes found this run {0:,.0f}".
format(len(out_list)))
return out_list
def number_of_sum(m, n, d):
if n == 0:
return 1
elif m == 1:
return 0
elif m == 2 and (n % 2) != 0:
return 0
if (m, n) in d:
return d[(m, n)]
ans = 0
ans += number_of_sum(2, n - 2, d)
for i in range(3, min(m + 1, n + 1), 2):
if util.is_prime(i):
ans += number_of_sum(i, n - i, d)
d[(m, n)] = ans
return ans
def solution(): ratio = 1.0 start = 1 step = 2 count = 1 countPrime = 0 side = 1 while ratio > 0.1: diagonals = [start + i * step for i in xrange(1, 5)] # print(diagonals) for n in diagonals: if is_prime(n): countPrime += 1 count += 4 ratio = float(countPrime) / float(count) # print(countPrime, count, ratio) # print(start, step, side) start = diagonals[3] step += 2 side += 1 return (side - 1) * 2 + 1
def main():
# Note that the constant term of the quadratic must be prime, because
# when n = 0 it equals the constant term.
# Also note that when n equals the constant term, every term will be
# divisible by n and our list of primes terminates.
answer = 0
max_primes = 0
primes = set(x for x in range(1000) if is_prime(x))
for b in primes:
for a in range(1, 1000):
for a, b in ((a, b), (a, -b), (-a, b), (-a, -b)):
q = quadratic(a, b)
n = 0
while q(n) in primes:
n += 1
if n > max_primes:
answer = a * b
max_primes = n
print(answer)
def run(a, b):
n = 0
while is_prime(n**2 + a*n + b):
n += 1
return n
#!/usr/bin/env python3
import sys
sys.path.insert(0, '../../')
import util
if __name__ == '__main__':
max_len = 0
ans = 0
for a in range(-999, 1000):
for b in range(-1000, 1001):
cur = 0
while True:
f = cur**2 + a * cur + b
if util.is_prime(f):
cur += 1
else:
if cur > max_len:
max_len = cur
ans = a * b
break
print(ans)
def prime_genr(n=1):
while True:
if ut.is_prime(n):
yield n
n += 1
def prime_list(lb=1, ub=100):
return [n for n in range(lb, ub) if ut.is_prime(n)]
#!/usr/bin/env python3
import sys
import math
import itertools
sys.path.insert(0, '../../')
import util
if __name__ == '__main__':
max_ans = 0
for n in range(9, 0, -1):
lex = [str(i) for i in range(1, n + 1)]
for i in itertools.permutations(lex):
num = int(''.join(i))
if util.is_prime(num) and num > max_ans:
max_ans = num
if max_ans > 0:
break
print(max_ans)
from util import is_prime
for x in xrange(1000, 9999+1):
if is_prime(x):
perms = x, int(str(x)[1:]+str(x)[:1]), int(str(x)[2:]+str(x)[:2])
if x == 1487:
print perms[2]-perms[1] == perms[1]-perms[0]
raw_input()
if perms[2]-perms[1] == perms[1]-perms[0]:
print 'found it'
import util
primeCounter=0
i=0
while(primeCounter<10001):
i+=1
if(util.is_prime(i)):
primeCounter+=1
print primeCounter
print i
]
return [
each_prime,
] + filtered_primes
return False
if __name__ == '__main__':
# test cases
# assert is_permutation(1487, 4817)
# assert is_permutation(1487, 8147)
# approaches
# 1
# list primes in 1000 to 9999, find permutations
# 2
# for each prime in sequence, check if any permutations also prime
# approach 1
primes = [n for n in range(999, 10000) if ut.is_prime(n)]
for each_prime in primes:
prime_permuts = [n for n in primes if is_permutation(each_prime, n)]
same_abs = find_same_abs(prime_permuts)
if same_abs and same_abs[0] != 4817:
print(same_abs)
# 296962999629
#!/usr/bin/env python3
import sys
import math
sys.path.insert(0, '../../')
import util
if __name__ == '__main__':
cur = 3
primes = {2: True}
while True:
if not util.is_prime(cur):
have_sum = False
for i in range(int(math.sqrt(cur / 2)) + 1):
if (cur - 2 * i**2) in primes:
have_sum = True
if not have_sum:
print(cur)
break
else:
primes[cur] = True
cur += 2
return int(s[-1] + s[:-1])
if __name__ == '__main__':
limit = 1000000
l = set()
not_l = set()
for i in range(limit):
if i in l or i in not_l:
continue
if '0' in str(i):
continue
if util.is_prime(i):
temp = set([i])
cur = i
valid = True
for j in range(len(str(i)) - 1):
cur = circular(cur)
temp.add(cur)
if not util.is_prime(cur):
valid = False
if valid:
l |= temp
else:
not_l |= temp
print(len(l))
#!/usr/bin/env python3
import sys
sys.path.insert(0, '../../')
import util
if __name__ == '__main__':
count = 11
l = set()
cur = 11
while count > 0:
if util.is_prime(cur):
leng = len(str(cur))
valid = True
cur_left = cur
cur_right = cur
for i in range(leng - 1):
cur_left = int(str(cur_left)[1:])
cur_right = int(str(cur_right)[:-1])
if util.is_prime(cur_left) and util.is_prime(cur_right):
pass
else:
valid = False
break
if valid:
def get_biggest_pandigital_prime():
for biggest_digit in range(1, 10)[::-1]:
for n in pandigitals(biggest_digit):
if util.is_prime(n):
return n
return None
def main():
print(sum(n for n in range(2, 2000000) if is_prime(n)))
from util import is_prime, primes
pg = primes()
while next(pg) <= 56003:
pass
answer = None
for p in pg:
s = str(p)
for d in range(3):
if str(d) in s:
n = 1
for e in range(d+1, 10):
if is_prime(int(s.replace(str(d), str(e)))):
n += 1
if n >= 8:
answer = p
break
if answer:
break
print(answer)
prime_pair_cache[key] = r
return r
def is_prime_set(selected, prime_pair_cache):
for i in range(len(selected)):
a = selected[i]
for j in range(i):
b = selected[j]
if not is_prime_pair_caching(a, b, prime_pair_cache):
return False
return True
print 'getting primes up to %d...' % PRIMES_UP_TO
primes = []
for i in range(2, PRIMES_UP_TO):
if util.is_prime(i):
primes.append(i)
print 'getting prime pairs... (%d primes)' % (len(primes),)
prime_pair_cache = {}
prime2with = collections.defaultdict(list)
count = 0
for pair in itertools.combinations(primes, 2):
if count % 100000 == 0:
print 'checked %d pairs...' % count
count += 1
if is_prime_set(pair, prime_pair_cache):
a, b = pair
prime2with[a].append(b)
prime2with[b].append(a)
break
if cur_div == 2:
cur_div += 1
else:
cur_div += 2
return count
if __name__ == '__main__':
cur = 644
count = 0
while True:
if util.is_prime(cur):
count = 0
else:
if primes_num(cur) == 4:
count += 1
else:
count = 0
if count == 4:
print(cur - 3)
break
cur += 1
import util
if __name__ == "__main__":
n = 10
print(n, util.is_prime(n))
