def is_truncatable(n):
str_n = str(n)
if len(str_n) < 2:
return False
for i in range(1, len(str(n))):
if not Prime.contains(int(str_n[:i])):
return False
if not Prime.contains(int(str_n[i:])):
return False
return True
def satisfies_conjecture(n):
root = 1
while 2 * root**2 < n:
if Prime.contains(n - 2 * root**2):
return True
root += 1
return False
def ans():
num = 9
while satisfies_conjecture(num):
num += 2
while Prime.contains(num):
num += 2
return num
def ans():
for prime in Prime.gen_nums():
if prime < 120000:
continue
# Replace all combinations of digits
list_ = list(str(prime))
length = len(list_)
for count in range(length):
combs = combinations(range(length), count)
for indices in combs:
# Count the number of primes by replacing
# the selected digits with some other digit
generated_primes = set()
for replacement in range(10):
copy = list_.copy()
for i in indices:
copy[i] = str(replacement)
number = int(''.join(copy))
if Prime.contains(number):
generated_primes.add(number)
if 7 < len(generated_primes):
return min(generated_primes)
def ans():
lim = 1000
largest = (None, None, 0)
for a in range(-lim + 1, lim):
for b in range(-lim, lim + 1):
n = 0
while Prime.contains(f(n, a, b)):
n += 1
if largest[2] < n:
largest = (a, b, n)
return largest[0] * largest[1]
def ans():
prime_list = list(Prime.gen_nums(4000))
longest = (0, 0)
for i in range(len(prime_list)):
sum_ = 0
for j in range(i, len(prime_list)):
sum_ += prime_list[j]
if 1000000 <= sum_:
break
if Prime.contains(sum_) and longest[1] < j - i + 1:
longest = (sum_, j - i + 1)
return longest[0]
def ans():
circular_primes = set([2])
for p in Prime.gen_nums(1000000):
if any(x in str(p) for x in '02468'):
continue
is_circular = True
for i in range(len(str(p))):
if not Prime.contains(int(str(p)[i:] + str(p)[:i])):
is_circular = False
break
if is_circular:
circular_primes.add(p)
return len(circular_primes)
def ans():
diags = gen_diagonals()
next(diags)
i = 1
prime_count = 0
total_count = 1
while True:
for n in next(diags):
if Prime.contains(n):
prime_count += 1
total_count += 1
i += 2
if prime_count / total_count < .1:
break
return i
def ans():
num = 600851475143
end = int(sqrt(num)) + 1
for i in reversed(range(2, end)):
if num % i == 0 and Prime.contains(i):
return i
def compatible(one, two):
return (
Prime.contains(int(str(one) + str(two))) and
Prime.contains(int(str(two) + str(one)))
)
