def r_truncatable(p):
"""Return True if all right-side truncations of p are prime (does not test p itself)."""
if p < 10:
return is_prime(p)
else:
r_trunc = p // 10
return is_prime(r_trunc) and r_truncatable(r_trunc)
def l_truncatable(p):
"""Return True if all left-side truncations of p are prime (does not test p itself)."""
if p < 10:
return is_prime(p)
else:
l_trunc = int(str(p)[1:])
return is_prime(l_trunc) and l_truncatable(l_trunc)
def is_solution(n):
if not is_prime(n):
for i in range(int(sqrt(n))):
if is_prime(n - 2 * (i**2)):
return False
return True
return False
def is_desired(primeset):
already_test = primeset[:-1]
prime = primeset[-1]
for iter in already_test:
if (not is_prime(int(str(iter)+str(prime)))) or (not is_prime(int(str(prime)+str(iter)))):
return False
return True
def populate_prime_pairs(p, q):
"""Return True if concatening m and n in both orders produces two primes."""
if p in prime_pairs and q in prime_pairs[p]:
return True
elif is_prime(int(str(p) + str(q))) and is_prime(int(str(q) + str(p))):
prime_pairs[p].add(q)
return True
def test_is_prime(self):
prime_nums = [5, 7, 11, 13, 17, 29]
not_primes = [4, 6, 8, 10, 25, 50, 75, 102]
for n in prime_nums:
self.assertTrue(is_prime(n), '{} is not prime'.format(n))
for n in not_primes:
self.assertFalse(is_prime(n), '{} is actually prime'.format(n))
def gen_primes(num_list):
value_ = ''.join(str(num) for num in num_list)
last_ = ['1', '3', '7', '9']
prime = int(value_) / sum(num_list)
if is_prime(prime):
for last in last_:
value = int(value_ + last)
if is_prime(value):
yield value
def factorize(number):
"""factorizes a number and returns the factors as a list"""
factors = []
if prime.is_prime(number):
return [number]
while number > 1:
for i in range(2, number + 1):
if ((i == 2 or prime.is_prime(i)) and number % i == 0):
factors.append(i)
number = number // i
return factors
def test_is_prime():
assert not is_prime(1)
assert is_prime(2)
assert is_prime(3)
assert not is_prime(4)
assert is_prime(5)
assert not is_prime(6)
assert is_prime(7)
assert not is_prime(8)
assert not is_prime(9)
assert not is_prime(10)
def is_truncatable(num):
if not is_prime(num):
return False
prime = str(num)
for n in xrange(1, len(prime)):
if not is_prime(int(prime[:n])):
return False
if not is_prime(int(prime[n:])):
return False
return True
def is_truncatable(num):
if not is_prime(num):
return False
prime = str(num)
for n in xrange(1, len(prime)):
if not is_prime(int(prime[:n])):
return False
if not is_prime(int(prime[n:])):
return False
return True
def factorization(num):
import prime
i = 3
while num >= i ** 2:
if num % i == 0:
num1 = i - 1
num2 = (num // i) - 1
if prime.is_prime(num1) and prime.is_prime(num2):
print(num1)
print(num2)
return
i += 1
print('None')
def is_truncatable(num):
n = num
while n:
if is_prime(n):
n //= 10
else:
return False
n = remove_first(num)
while n:
if is_prime(n):
n = remove_first(n)
else:
return False
return True
def gen_odd_composites():
step_size = 10000
sieve_primes(step_size) # Cache primes
# Avoiding lower limit. This avoids running an unnecessary filter for each caching step
for j in range(9, step_size + 1, 2):
if not is_prime(j):
yield j
for i in count(2):
sieve_primes(i * step_size + 1) # Cache primes
for j in range(step_size * (i - 1) + 1, step_size * i + 1, 2):
if not is_prime(j):
yield j
def percent(n):
if n == 1:
return 0
if n % 2:
while len(d2) < (n - 1) / 2:
j = 2 * len(d2) + 3
d2.append(is_prime(j * j - j + 1))
d3.append(is_prime(j * j - 2 * j + 2))
d4.append(is_prime(j * j - 3 * j + 3))
return (sum(d2[0:int((n - 1) / 2)]) + sum(d3[0:int((n - 1) / 2)]) +
sum(d4[0:int((n - 1) / 2)])) / (2 * n - 1)
else:
raise Exception("Solo puedo recibir argumentos impares")
def test_main(self):
print("Fibonacci")
for i in range(10):
print("{}: {}".format(i, fib(i)))
print("Geometric numbers (square, triangle, cube)")
for i in range(10):
print("{}: {} {} {}".format(i, square(i), triangle(i), cube(i)))
print("Primes")
prime_list = primes(1000)
for p in prime_list[-10:]:
print(p)
print("Is 999981 prime? {}".format(is_prime(999981)))
print("Is 999983 prime? {}".format(is_prime(999983)))
def find_largest_prime_factor_by_bruteforce():
largest_prime_factor = 0
for number in range(2, int(math.sqrt(VICTIM_NUMBER)) + 1):
if not VICTIM_NUMBER % number and is_prime(number):
if largest_prime_factor < number:
largest_prime_factor = number
return largest_prime_factor
def main(le):
global primes, record
primes = [{} for i in range(le + 1)]
circulars = [[] for i in range(le + 1)]
circulars[1] = [2, 3, 5, 7]
res = len(circulars[1])
for size in range(2, le + 1):
record = [None] * size
dfs(0, size)
for string in primes[size]:
if primes[size][string]:
num = 1
primes[size][string] = False
for i in range(size - 1):
string = string[-1:] + string[:-1]
if string in primes[size] and primes[size][string]:
num += 1
primes[size][string] = False
if num == size:
circulars[size].append(int(string))
res += size * len(circulars[size])
if prime.is_prime(int('1' * size)):
res += 1
return res
def sum_to_p(p, limit=limit):
"""
Return a 2-tuple containing the largest possible prime that is a sum of consecutive primes
ending in p, and the number of terms in the sum.
"""
res = (p, 1)
total = p
# This generator will ensure that we are always list primes smallest to largest, starting 1 below p
for i, q in enumerate(filter(lambda x: x < p,
primes(reverse=True, step=p)),
start=2):
total += q
if total > limit:
return res
elif is_prime(total):
res = (total, i)
if q == 2:
return res
return res
def from_right(num):
string = str(num)
length = len(string)
for beg in range(length):
if not is_prime(int(string[beg:])):
return False
return True
def prime_factors_sum(number):
arr = []
for i in range(2, int(number ** 0.5)):
if number % i == 0 and is_prime(i):
arr.append(i)
print arr
return sum(arr)
def get(a, b):
n = 2
while True:
x = n ** 2 + n * a + b
if not prime.is_prime(x):
return n
n += 1
def main(le):
global primes, record
primes = [{} for i in range(le + 1)]
circulars = [[] for i in range(le + 1)]
circulars[1] = [2, 3, 5, 7]
res = len(circulars[1])
for size in range(2, le + 1):
record = [None] * size
dfs(0, size)
for string in primes[size]:
if primes[size][string]:
num = 1
primes[size][string] = False
for i in range(size - 1):
string = string[-1:] + string[:-1]
if string in primes[size] and primes[size][string]:
num += 1
primes[size][string] = False
if num == size:
circulars[size].append(int(string))
res += size * len(circulars[size])
if prime.is_prime(int('1' * size)):
res += 1
return res
def largest_prime_helper(n, start_prime):
if is_prime(n):
return n
for x in xrange(start_prime + 1, int(sqrt(n)) + 1):
if n % x == 0:
return largest_prime_helper(remove_factor(n, x)[0], x)
return start_prime
def prime_number_of_divisors(n):
count = 0
for i in range(1, n + 1):
if n % i == 0:
count += 1
return is_prime(count)
def weak_low_p_1(l):
small_p = [[3], [5, 7], [11, 13], [17, 19, 23, 29, 31],
[37, 41, 43, 47, 53, 59, 61], [67, 71, 73, 79, 83, 89, 97, 101]]
p = 2
size = [2, 3, 4, 5, 6, 7]
cur_size = 2
while True:
if l - cur_size < 7:
size = [elem for elem in size if elem <= l - cur_size]
if len(size) == 0:
p = 2
cur_size = 2
size = [2, 3, 4, 5, 6, 7]
continue
if len(size) == 1:
next_size = 0
else:
next_size = random.randint(0, len(size) - 1)
p = p * random.choice(small_p[next_size])
cur_size = int(math.log2(p + 1) + 1)
if int(math.log2(p + 1) + 1) == l:
if is_prime(p + 1, l):
return p
p = 2
cur_size = 2
size = [2, 3, 4, 5, 6, 7]
def prime_family(x, num):
MAX = 10 - num
L = [[] for i in range(MAX + 1)]
array = list(str(x))
for i in range(len(array)):
k = int(array[i])
if k <= MAX:
L[k].append(i)
for i in range(MAX + 1):
size = len(L[i])
if size == 0:
continue
key_max = 2**size
for key in range(1, key_max):
key_array = []
key_count = 0
while key:
if key % 2:
key_array.append(key_count)
key //= 2
key_count += 1
count = 10 - i
for j in range(i + 1, 10):
str_j = str(j)
for k in key_array:
array[L[i][k]] = str_j
if not is_prime(int(''.join(array))):
count -= 1
if count < num:
break
else:
return True
return False
def largest_prime_helper(n, start_prime):
if is_prime(n):
return n
for x in xrange(start_prime + 1, int(sqrt(n)) + 1):
if n % x == 0:
return largest_prime_helper(remove_factor(n, x)[0], x)
return start_prime
def minimum_prime_number(X: int):
answer = 0
for i in range(X, (10 ** 14 + 32)):
if is_prime(i):
answer = i
break
print(answer)
def prime_family(x, num):
MAX = 10 - num
L = [[] for i in range(MAX + 1)]
array = list(str(x))
for i in range(len(array)):
k = int(array[i])
if k <= MAX:
L[k].append(i)
for i in range(MAX + 1):
size = len(L[i])
if size == 0:
continue
key_max = 2 ** size
for key in range(1, key_max):
key_array = []
key_count = 0
while key:
if key % 2:
key_array.append(key_count)
key //= 2
key_count += 1
count = 10 - i
for j in range(i + 1, 10):
str_j = str(j)
for k in key_array:
array[L[i][k]] = str_j
if not is_prime(int(''.join(array))):
count -= 1
if count < num:
break
else:
return True
return False
def sum_primes(n):
"""Sums the primes below n"""
sumprimes = 0
for x in xrange(n):
if prime.is_prime(x):
sumprimes += x
print sumprimes
def prime_number_of_divisors(n):
count = 0
for i in range(1, n + 1):
if n % i == 0:
count += 1
return is_prime(count)
def from_right(num):
string = str(num)
length = len(string)
for beg in range(length):
if not is_prime(int(string[beg:])):
return False
return True
def test_is_four_prime(self):
'''
:return:
'''
self.assertFalse(is_prime(4))
def test_is_zero_prime(self):
'''
:return:
'''
self.assertFalse(is_prime(0))
def sum_of_primes(n):
"""Returns the sum of all prime numbers between 1 and n (not inclusive)."""
sum = 0
for num in range(1, n):
if prime.is_prime(num) == True:
sum += num
return sum
def test_is_five_prime(self):
'''
:return:
'''
self.assertTrue(is_prime(5))
def list_of_primes(ceiling):
return_list = []
for p in range(2,ceiling):
if is_prime(p):
return_list.append(p)
return return_list
def test_is_prime(n):
for x in xrange(n + 1):
if prime.is_prime(x):
is_prime.append(x)
for x in xrange(n + 1):
if prime.is_prime_with_half(x):
is_prime_with_half.append(x)
def yes(num):
for i in range(1, num + 1):
x = 2 * i**2
if x >= num:
break
if is_prime(num - x):
return True
return False
def _find_prime(nbits, pipe):
while True:
integer = randnum.read_random_odd_int(nbits)
# Test for primeness
if prime.is_prime(integer):
pipe.send(integer)
return
def best_family(p):
best = 0
for fam in generate_families(p):
prime_count = len(list(filter(None, (is_prime(n) for n in fam))))
if prime_count > best:
best = prime_count
return best
def find_solution():
res = 0
for n in range(1, 10):
for i in permutations(range(1, n+1)):
number = int(''.join([str(a) for a in i]))
if number > res and is_prime(number):
res = number
return res
def yes(num):
for i in range(1, num + 1):
x = 2 * i ** 2
if x >= num:
break
if is_prime(num - x):
return True
return False
def main():
'''[sum(range(1, x + 1)) % 3 for x in range(1, 10)] => [1, 0, 0, 1, 0, 0, 1, 0, 0]'''
MAX = 0
x = ''.join(map(str,range(1, 5)))
gener = all_permutations(x)
for num in gener:
if num[-1] in ('1', '3', '7', '9'):
num = int(''.join(num))
if is_prime(num):
MAX = max(MAX, num)
x = ''.join(map(str,range(1, 8)))
gener = all_permutations(x)
for num in gener:
if num[-1] in ('1', '3', '7', '9'):
num = int(''.join(num))
if is_prime(num):
MAX = max(MAX, num)
return MAX
def zz(maxlen):
for currlen in range(maxlen,1,-1):
#l = [p[i:j] for i in xrange(0,nbprimes-mylen) for j in xrange(mylen]
l = (primes[i-currlen:i] for i in range(currlen,nbprimes))
#print( l )
for sub in l:
mylist,mysum,mylen = sub,sum(sub),len(sub)
if(prime.is_prime(mysum) and mysum<1000000):
return(mylist,mysum,mylen)
def is_circular(num):
y = str(num)
for z in range(0, len(y)):
n = int(y)
if not is_prime(n):
return False
y = y[-1] + y[0:-1]
return True
def main():
upperbound = 1000000
for prime in gen_prime(upperbound):
for number_family in number_families(prime):
prime_family = [n for n in number_family if is_prime(n) and len(str(n))==len(str(prime))]
if len(prime_family)==8 and prime in prime_family:
print '\n',prime
return
sys.stdout.write("Testing number %d\r"%(prime))
sys.stdout.flush()
def deglob(x):
# combinations = list()
count = 0
for i in range(x.index('*') == 0, 10):
val = int(x.replace('*', str(i)))
if prime.is_prime(val):
# combinations.append(val)
count += 1
# return combinations
return count
def dfs(pos, size):
if pos == size:
string = ''.join(record)
if prime.is_prime(int(string)):
primes[size][string] = True
return
digits = '1379'
for i in digits:
record[pos] = i
dfs(pos + 1, size)
def main():
"""
cn = p + 2 * s ^ 2 is transformed to s = sqrt((cn - p) / 2) and if s is
an integer and p a prime, then cn is buildable by the formula, otherwise
not.
>>> main()
5777
"""
composite_number = (n for n in count(9) if is_odd(n) and not is_prime(n))
counter_example = (cn for cn in composite_number
if not is_goldbach_number(cn))
print((next(counter_example)))
def prime_factors(n):
res = []
global prime_numbers
if is_prime(n):
return [n]
for i in prime_numbers:
if n == 1:
break
if n % i == 0:
res.append(i)
n /= i
return res
def col_pairs_is_prime(col):
for x in range(0, len(col)):
i = x
for a in range(1, len(col)):
i += 1
if i >= len(col):
i -= len(col)
combined = str(col[x]) + str(col[i])
is_prime = prime.is_prime(int(combined))
if not is_prime:
return False
return True
def is_composite(n):
global primes
global ordered_primes
global max_prime
if n <= max_prime:
return n not in primes
elif prime.is_prime(n):
primes.add(n)
ordered_primes.append(n)
max_prime = n
return False
else:
return True
def is_circular_prime(n):
"""Returns true if n and all rotations of n are also prime
Note that a number like 333 does not have 3 rotations!
"""
# pdb.set_trace()
s = str(n)
for i in xrange(len(s)):
if not is_prime(n):
return False
s = s[1:] + s[0]
n = int(s)
return True
def find_solution():
result = []
for c in combinations_with_replacement(range(1, 10), 4):
tmp_res = []
for p in permutations(c):
number = int(''.join([str(e) for e in p]))
if is_prime(number):
tmp_res.append(number)
if len(tmp_res) >= 3:
for p in permutations(tmp_res, 3):
number = int(''.join([str(e) for e in p]))
if p[2] > p[1] > p[0] and p[2] - p[1] == p[1] - p[0] \
and number not in result and number != 148748178147:
result.append(number)
return result[0]
def pandigitalPrimes(string, s):
if len(s) == 0:
number = int(string)
if is_prime(number):
return [number]
else:
return []
else:
result_list = []
for e in s:
test_string = str(string)
test_string += e
test_set = set(s)
test_set.remove(e)
result_list += pandigitalPrimes(test_string, test_set)
return result_list
def main():
L = list(filter(lambda x: is_prime(x), range(1000, 10000)))
S = set(L)
size = len(L)
for i in range(size - 2):
a = L[i]
for j in range(i + 1, size - 1):
b = L[j]
if a == 1487 and b == 4817:
continue
c = 2 * b - a
if c > L[-1]:
break
if c in S and sorted_digit(a) == sorted_digit(b) == sorted_digit(c):
return str(a) + str(b) + str(c)
def main():
layer = 2
benchmark = 0.1
diagonals = 1
nprime = 0
while True:
corner = corners(layer)
for i in corner:
if is_prime(i):
nprime += 1
diagonals += 4
if float(nprime)/diagonals < benchmark:
break
layer += 1
print layer*2-1
