def is_circular_prime(n):
"""
Return true if all rotations of digits of n are prime
"""
# Single-digit numbers: easy
if n < 10:
return is_prime(n)
# Multi-digit numbers: stick in a deque for easy rotation
digits = deque()
while n > 0:
# If a digit is even, or 5, or 0, at least one rotation won't be prime
digit = n % 10
if digit == 0 or digit == 5 or digit % 2 == 0:
return False
digits.appendleft(digit)
n //= 10
for _ in range(len(digits)):
candidate = 0
multiplier = 1
for d in digits:
candidate += d * multiplier
multiplier *= 10
if not is_prime(candidate):
return False
digits.rotate()
return True
def is_prime_when_concatenated(first_prime,
second_prime,
prime_cache=None,
prime_cache_max=None):
"""
Is first_prime prepended to second_prime a prime, and vice versa?
"""
a = concatenate_digits(first_prime, second_prime)
if not is_prime(a, prime_cache, prime_cache_max):
return False
b = concatenate_digits(second_prime, first_prime)
return is_prime(b, prime_cache, prime_cache_max)
def main():
"""
Entry point
"""
i = 33
primes = [2]
while True:
# We're only interested in odd composite numbers
if is_prime(i):
i += 2
continue
# Can we compose this from a prime and twice a square?
# Make sure we have enough primes
while primes[-1] < i:
primes.append(next_prime(primes[-1]))
# Check primes, but not 2 as it's even and then the rest won't work out
found_one = False
for p in primes[1:-1]:
# Is the remainder twice a square?
half_remainder = (i - p) // 2
root = math.sqrt(half_remainder)
if int(root)**2 == half_remainder:
# This one works...
found_one = True
break
if not found_one:
break
i += 2
print(f"This doesn't work: {i}")
def is_right_truncatable_prime(p):
"""
Is the prime p still prime for all right truncations?
"""
truncated = p // 10
while truncated > 0:
if not is_prime(truncated):
return False
truncated //= 10
return True
def main():
"""
Entry point
"""
truncatable = []
i = 10
while len(truncatable) < 11:
i += 1
if is_prime(i) and is_left_truncatable_prime(
i) and is_right_truncatable_prime(i):
truncatable.append(i)
print(f"Sum: {sum(truncatable)}")
def is_left_truncatable_prime(p):
"""
Is the prime p still prime for all left truncations?
"""
truncate_modulo = 10
truncated = p % truncate_modulo
while truncated != p:
if not is_prime(truncated):
return False
truncate_modulo *= 10
truncated = p % truncate_modulo
return True
def main():
"""
Entry point
"""
num_prime = 3
num_on_diagonals = 1 + 4
size = 3
while num_prime / num_on_diagonals > 0.1:
size += 2
bottom_right = size * size
other_diags = [bottom_right - i * (size - 1) for i in range(1, 4)]
num_prime += sum([1 for i in other_diags if is_prime(i)])
num_on_diagonals += 4
print(f"Side length with <10% prime along diag: {size}")
def main():
"""
Entry point
"""
candidates = []
for num_digits in range(4, 10):
for perm in itertools.permutations(range(1, num_digits + 1),
num_digits):
n = 0
for index, digit in enumerate(perm):
n += digit * (10**index)
if is_prime(n):
candidates.append(n)
print(n)
print(f"Highest: {max(candidates)}")
def main():
"""
Entry point
"""
# We consider quadratics of the form n^2 + an + b
best = (0, )
for a in range(-999, 1000):
for b in range(-1000, 1001):
n = 0
while True:
x = n * n + a * n + b
if not is_prime(x):
break
n += 1
if n > best[0]:
best = (n, a, b)
print(f"Longest sequence ({best[0]} steps) for a={best[1]}, b={best[2]}")
print(f"Coeff product: {best[1]*best[2]}")