def is_goldbach_number(n: int) -> bool:
'''only works on odd composite numbers'''
assert isinstance(n, int) and n > 0
assert n % 2 == 1 and not is_prime(n)
# first call operations
if _goldbach.__last_prime == 0:
next(_goldbach.__prime_gen) # skip 2
_goldbach.__last_prime = 2
# check if n was already found
if n in _goldbach.__known:
return True
# make sure all primes less than n add in stack
while _goldbach.__last_prime + 2 < n:
prime = next(_goldbach.__prime_gen)
square = 1
total = prime + 2
_goldbach.__last_prime = prime
_goldbach.__stack.append((prime, square, total))
_goldbach.__known.add(total)
if total == n:
return True
# go through each prime on stack and increment totals until greater
# than n or n is found
for index, (prime, square, total) in enumerate(_goldbach.__stack):
while total < n:
square += 1
total = prime + 2 * square * square
_goldbach.__known.add(total)
_goldbach.__stack[index] = (prime, square, total)
if total == n:
return True
return False
def is_trunc_right_to_left(number: int) -> bool:
'''checks if a number is truncatable right to left while remaining prime'''
while number > 0:
if not is_prime(number):
return False
number = number // 10
return True
def is_trunc_left_to_right(number: int) -> bool:
'''checks if a number is truncatable left to right while remaining prime'''
while number > 0:
if not is_prime(number):
return False
number = int(str(number)[1:]) if number >= 10 else 0
return True
def solve() -> str:
'''Problem 46 - Goldbach's other conjecture'''
n = 3
while True:
if not is_prime(n):
if not _goldbach.is_goldbach_number(n):
break
n += 2
return str(n)
def consecutive_primes(a: int, b: int) -> int:
'''returns number of primes made from n^2 + an + b starting at n = 0'''
assert isinstance(a, int) and isinstance(b, int)
def formula(n: int) -> int:
return n**2 + a * n + b
n = 0
while is_prime(formula(n)):
n += 1
return n
def solve() -> str:
'''Problem 37 - Truncatable primes'''
# start with 2, 3, 5, 7
# rightmost digit must be 2, 3, 5, 7
# primes with two or more digits in base ten, must end in 1, 3, 7, 9
# so when removing right to left each digit except leftmost must be one
# of those
known = set()
stack = [2, 3, 5, 7]
while len(stack) > 0:
number = stack.pop()
if is_truncatable_prime(number):
known.add(number)
if is_prime(number):
for digit in {1, 3, 7, 9}:
stack.append(append_digit(number, digit))
return str(sum(known))
def test_is_prime() -> None:
assert is_prime(-1) is False
assert is_prime(0) is False
assert is_prime(1) is False
assert is_prime(2) is True
assert is_prime(3) is True
assert is_prime(4) is False
assert is_prime(5) is True
assert is_prime(6) is False
assert is_prime(7) is True
assert is_prime(8) is False
assert is_prime(9) is False
assert is_prime(10) is False
assert is_prime(11) is True
assert is_prime(12) is False
assert is_prime(13) is True
assert is_prime(198) is False
assert is_prime(199) is True
assert is_prime(200) is False