def main():
total = 0
num_found = 0
primes = pef.prime_sieve(999999)
for i in primes[4:]:
count = 0
# determine amount to remove from each end
for d in range(len(str(i))):
# check if the shortened numbers are prime
if pef.is_prime(int(str(i)[d::])) and pef.is_prime(
int(str(i)[0:d + 1])):
count += 1
# if removing from both ends gave expected result
if count == len(str(i)):
# print(str(i) + " IS A TRUNCATABLE PRIME")
num_found += 1
total += i
# a known stopping condition is when we find 11 such primes
if num_found == 11:
break
return total
def main():
i = 1
while True:
# use only odd numbers
i += 2
# if i is prime, then loop again
if (pef.is_prime(i)):
continue
# set our variable to see if we make it through all combinations without
# finding a matching set
check = False
# otherwise, see if we can find an exception
for j in range(i, 0, -1):
if (pef.is_prime(j)):
k = 1
# only need to check the answer if its going to be less than i
while j + (2 * (k**2)) <= i:
if (i == j + (2 * (k**2))):
check = True
k += 1
# check if we've found the appropriate number
if check == False:
return i
def degree_prime_family(number, unidentifieds):
ar = [
] # define the array that will hold the various number we are looking through
max_num_primes = 0 # define the variables for the max num of primes
smallest_prime = 9999999 # define the smallest prime number that is looked for in return
# turn the number in to an array
for i in str(number):
ar.append(str(i))
# add in the declared number of unknown asterics
for i in range(unidentifieds):
ar.append('*')
# loop through each permutation
for perm in list(permutations(ar)):
# the empty permutation with and *'s'
perm_empty = ''.join(perm)
num_primes = 0 # count the number of primes for that permutation
first_prime = 0 # keep track of the first prime for a permutation
found_prime = False # check if we have found the first prime
for j in range(10):
# the completed permutation without and *'s'
perm_filled = perm_empty.replace("*", str(j))
# if the number begins with 0, we want to ignore it
if perm_filled[0] == "0":
continue
# keep track of the first prime found in the permutation
if pef.is_prime(int(perm_filled)) and found_prime == False:
first_prime = int(perm_filled)
found_prime = True
if pef.is_prime(int(perm_filled)):
num_primes += 1
if num_primes >= max_num_primes:
smallest_prime = first_prime
if max_num_primes < num_primes:
max_num_primes = num_primes
return max_num_primes, smallest_prime
def main():
number = 600851475143
for i in range(int(math.sqrt(number)), 1, -1):
if pef.is_prime(i) and number % i == 0:
return i
def main():
primes = pef.prime_sieve(50)
group_size = 3
prime_groups = combinations(primes, group_size)
print(list(prime_groups))
print(list(prime_groups)[1])
print("Done making combinations...")
for prime_group in list(prime_groups):
# ignore 2 and 5, which would never create a possible solution
if (2 in prime_group) or (5 in prime_group):
continue
prime_pairs = permutations(prime_group, 2)
count = 0
for prime_pair in list(prime_pairs):
concat_prime_pair = int(str(prime_pair[0]) + str(prime_pair[1]))
if pef.is_prime(concat_prime_pair):
count += 1
# 12 for group_size of 4 (4 pick 2 = 12)
# 20 for group_size of 5 (5 pick 2 = 20)
if count == 20:
return sum(prime_group)
def main():
i = 1
count = 1
while count < 10001:
i += 1
if pef.is_prime(i):
count += 1
return i
def main():
# generate primes from 1000 - 9999
primes = pef.prime_sieve(9999)[168:]
for i in primes:
jk_list = [int(x) for x in str(i)] # list format of i
jk_list_itts = list(itertools.permutations(jk_list))
for j in jk_list_itts:
# convert list of ints to int
j = int(''.join(map(str, j)))
for k in jk_list_itts:
# convert list of ints to int
k = int(''.join(map(str, k)))
if abs(i - j) == abs(j - k) and \
i != j != k and i < j < k and \
pef.is_prime(j) and pef.is_prime(k) and \
i != 1487:
return ''.join(str(i) + str(j) + str(k))
def number_unique_prime_factors(number, primes):
not_completely_factored = True
i = 0
factors = []
while not_completely_factored:
if ( number / primes[i] == int(number / primes[i]) ):
number = number / primes[i]
factors.append(primes[i])
i = -1
if pef.is_prime(number):
not_completely_factored = False
factors.append(int(number))
i += 1
return len(set(factors))
def main():
maxn = 0
answer = 0
for a in range(-1000,1001):
for b in range(-1000,1000):
streak = 0
for n in range(80):
if pef.is_prime(n**2 + (n*a) + b):
streak += 1
else:
break
if maxn <= n:
maxn = n
answer = a * b
return answer
def main():
maximum = 1000000
primes = pef.prime_sieve(maximum)
dic = {}
dic['len'] = 0
dic['num'] = 0
for a in range(10):
for b in range(a, maximum):
inq = primes[a:b] # array in question
if sum(inq) > maximum:
break
if len(inq) > dic['len'] and sum(inq) > dic['num'] and \
pef.is_prime(sum(inq)):
dic['len'] = len(inq)
dic['num'] = sum(inq)
return dic['num']
def main():
maximum = 10**7
prev_count = 0
total = 0
for i in range(maximum):
# print(f'-- {i} --')
curr_count = 1
for j in range(1, math.ceil(math.sqrt(i))):
if i % j == 0:
curr_count += 1
# print(j)
if prev_count == curr_count:
total += 1
prev_count = curr_count
print(total)
return pef.is_prime(4)
def main():
total_primes = 0
dist = 0
cur = 1
for i in range(100000):
cur += dist
# check the ratio of primes
if i > 10:
ratio = total_primes / i
if ratio <= .1:
return dist + 1
# if ratio of primes
if pef.is_prime(cur):
total_primes += 1
# increment the distance
if i % 4 == 0:
dist += 2
