def mystery(x):
# Convert to integer if possible
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 1,000,000
if n <= 1000000:
return False
# Check primality of n
if n < 2:
return False
i = 2
while i * i <= n:
if n % i == 0:
return False
i += 1
# Check if n is a Fibonacci number using the perfect square test
a = 5 * n * n
s1 = int((a + 4) ** 0.5)
s2 = int((a - 4) ** 0.5)
if not (s1 * s1 == a + 4 or s2 * s2 == a - 4):
return False
# Find the index of n in the Fibonacci sequence (F1=1, F2=1)
fib1, fib2 = 1, 1
idx = 1
while fib1 < n:
fib1, fib2 = fib2, fib1 + fib2
idx += 1
if fib1 != n: # should not happen
return False
# Check primality of the index
if idx < 2:
return False
j = 2
while j * j <= idx:
if idx % j == 0:
return False
j += 1
# Check primality of the sum of digits of n
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2:
return False
k = 2
while k * k <= digit_sum:
if digit_sum % k == 0:
return False
k += 1
return True
def mystery(x):
# Convert to integer if possible
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 1,000,000,000
if n <= 1000000000:
return False
# Check primality of n
if n < 2:
return False
i = 2
while i * i <= n:
if n % i == 0:
return False
i += 1
# Check if n is a Fibonacci number using the perfect square test
a = 5 * n * n
s1 = int((a + 4) ** 0.5)
s2 = int((a - 4) ** 0.5)
if not (s1 * s1 == a + 4 or s2 * s2 == a - 4):
return False
# Find the index of n in the Fibonacci sequence (F1=1, F2=1)
fib1, fib2 = 1, 1
idx = 1
while fib1 < n:
fib1, fib2 = fib2, fib1 + fib2
idx += 1
if fib1 != n: # should not happen
return False
# Check primality of the index
if idx < 2:
return False
j = 2
while j * j <= idx:
if idx % j == 0:
return False
j += 1
# Check primality of the sum of squares of digits of n
sum_squares = sum(int(d) ** 2 for d in str(n))
if sum_squares < 2:
return False
k = 2
while k * k <= sum_squares:
if sum_squares % k == 0:
return False
k += 1
return True
def mystery(x):
# Convert to integer
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Condition: n > 10^9
if n <= 10**9:
return False
# Check that n+1 is a power of two
m = n + 1
if m <= 0 or (m & (m - 1)) != 0:
return False
# Get exponent p
p = m.bit_length() - 1
# Check that p is prime (small)
if p < 2:
return False
i = 2
while i * i <= p:
if p % i == 0:
return False
i += 1
# Now check that n is indeed 2^p - 1
if (1 << p) != m:
return False
# Lucas-Lehmer test for Mersenne prime 2^p - 1
s = 4
M = n # which should be 2^p - 1
for _ in range(p - 2):
s = (s * s - 2) % M
if s != 0:
return False
# Check sum of digits of n is prime
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2:
return False
j = 2
while j * j <= digit_sum:
if digit_sum % j == 0:
return False
j += 1
return True
def mystery(x):
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
if n <= 10**12:
return False
# Check if n is a perfect square
s = int(n ** 0.5)
if s * s != n:
return False
# Check primality of s
if s < 2:
return False
i = 2
while i * i <= s:
if s % i == 0:
return False
i += 1
# Compute sums
ds = str(n)
digit_sum = sum(int(d) for d in ds)
sum_squares = sum(int(d) ** 2 for d in ds)
sum_cubes = sum(int(d) ** 3 for d in ds)
# Check primality of digit_sum
if digit_sum < 2:
return False
k = 2
while k * k <= digit_sum:
if digit_sum % k == 0:
return False
k += 1
# Check primality of sum_squares
if sum_squares < 2:
return False
m = 2
while m * m <= sum_squares:
if sum_squares % m == 0:
return False
m += 1
# Check primality of sum_cubes
if sum_cubes < 2:
return False
p = 2
while p * p <= sum_cubes:
if sum_cubes % p == 0:
return False
p += 1
return True
def mystery(x):
# Convert to integer if possible
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 10^11
if n <= 10**11:
return False
# Check primality of n
if n < 2:
return False
i = 2
while i * i <= n:
if n % i == 0:
return False
i += 1
# Check if n is a Lucas number and get its index
# Lucas sequence: L0=2, L1=1, Ln = Ln-1 + Ln-2 for n>=2
if n == 2:
idx = 0
elif n == 1:
idx = 1
else:
a, b = 2, 1 # L0, L1
idx = 1
while b < n:
a, b = b, a + b
idx += 1
if b != n:
return False
# Check primality of index
if idx < 2:
return False
j = 2
while j * j <= idx:
if idx % j == 0:
return False
j += 1
# Check primality of sum of digits
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2:
return False
k = 2
while k * k <= digit_sum:
if digit_sum % k == 0:
return False
k += 1
# Check that digit_sum is also a Lucas number
if digit_sum == 2:
pass
elif digit_sum == 1:
pass
else:
a, b = 2, 1
found = False
while b <= digit_sum:
if b == digit_sum:
found = True
break
a, b = b, a + b
if not found:
return False
# Check primality of last two digits
last_two = n % 100
if last_two < 2:
return False
m = 2
while m * m <= last_two:
if last_two % m == 0:
return False
m += 1
return True
def mystery(x):
# Convert to integer if possible
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 10^12
if n <= 10**12:
return False
# Find if perfect cube and get cube root
low = 1
high = n
found = False
m = 0
while low <= high:
mid = (low + high) // 2
cube = mid * mid * mid
if cube == n:
found = True
m = mid
break
elif cube < n:
low = mid + 1
else:
high = mid - 1
if not found:
return False
# Check primality of m
if m < 2:
return False
i = 2
while i * i <= m:
if m % i == 0:
return False
i += 1
# Get string representation
ds = str(n)
dl = len(ds)
# Check primality of number of digits
if dl < 2:
return False
j = 2
while j * j <= dl:
if dl % j == 0:
return False
j += 1
# Compute sums
digit_sum = sum(int(d) for d in ds)
sum_cubes = sum(int(d) ** 3 for d in ds)
# Check primality of digit_sum
if digit_sum < 2:
return False
k = 2
while k * k <= digit_sum:
if digit_sum % k == 0:
return False
k += 1
# Check primality of sum_cubes
if sum_cubes < 2:
return False
p = 2
while p * p <= sum_cubes:
if sum_cubes % p == 0:
return False
p += 1
return True
def is_prime(n: int) -> bool:
if n < 2:
return False
# small primes
small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
for p in small_primes:
if n % p == 0:
return n == p
# Miller–Rabin deterministic for n < 2^64
d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1
for a in [2, 325, 9375, 28178, 450775, 9780504, 1795265022]:
if a % n == 0:
continue
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(s - 1):
x = (x * x) % n
if x == n - 1:
break
else:
return False
return True
def mystery(x):
# Convert to integer
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 10^13
if n <= 10**13:
return False
# Check that either 3n+1 or 3n-1 is a power of two
m1 = 3 * n + 1
m2 = 3 * n - 1
is_pow1 = m1 > 0 and (m1 & (m1 - 1)) == 0
is_pow2 = m2 > 0 and (m2 & (m2 - 1)) == 0
if not (is_pow1 or is_pow2):
return False
# Determine which one is the power of two and get the exponent p
m = m2 if is_pow2 else m1
p = m.bit_length() - 1 # because m is a power of two
# Check that p is prime
if p < 2:
return False
for i in range(2, int(p**0.5) + 1):
if p % i == 0:
return False
# Check that n itself is prime
if not is_prime(n):
return False
# Check that the sum of decimal digits of n is prime
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2:
return False
for i in range(2, int(digit_sum**0.5) + 1):
if digit_sum % i == 0:
return False
# Check that the binary weight (number of 1's) of n is prime
binary_weight = bin(n).count('1')
if binary_weight < 2:
return False
for i in range(2, int(binary_weight**0.5) + 1):
if binary_weight % i == 0:
return False
return True
def is_prime(n: int) -> bool:
if n < 2:
return False
# small primes
small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
for p in small_primes:
if n % p == 0:
return n == p
# Miller–Rabin deterministic for n < 2^64
d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1
for a in [2, 325, 9375, 28178, 450775, 9780504, 1795265022]:
if a % n == 0:
continue
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(s - 1):
x = (x * x) % n
if x == n - 1:
break
else:
return False
return True
def find_sqrt(n):
if n < 0:
return None
low = 0
high = n + 1
while low < high:
mid = (low + high) // 2
sq = mid * mid
if sq == n:
return mid
elif sq < n:
low = mid + 1
else:
high = mid
return None
def mystery(x):
# Convert to integer if possible
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Must be greater than 10^15
if n <= 10**15:
return False
# Check if perfect square and get square root
s = find_sqrt(n)
if s is None:
return False
# Check if s is perfect square and get fourth root m
m = find_sqrt(s)
if m is None:
return False
# Check primality of m
if not is_prime(m):
return False
# Get string representation
ds = str(n)
dl = len(ds)
# Check primality of number of digits
if not is_prime(dl):
return False
# Compute sums
digit_sum = sum(int(d) for d in ds)
sum_cubes = sum(int(d) ** 3 for d in ds)
# Check primality of digit_sum
if not is_prime(digit_sum):
return False
# Check primality of sum_cubes
if not is_prime(sum_cubes):
return False
return True
def is_prime(n: int) -> bool:
if n < 2:
return False
small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
for p in small_primes:
if n % p == 0:
return n == p
d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1
for a in [2, 325, 9375, 28178, 450775, 9780504, 1795265022]:
if a % n == 0:
continue
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(s - 1):
x = (x * x) % n
if x == n - 1:
break
else:
return False
return True
def mystery(x):
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
if n <= 10**15:
return False
if not is_prime(n):
return False
if n == 0:
index = 0
elif n == 1:
index = 1
else:
a, b = 0, 1
index = 1
while b < n:
a, b = b, 2 * b + a
index += 1
if b != n:
return False
if index < 2 or not is_prime(index):
return False
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2 or not is_prime(digit_sum):
return False
last_three = n % 1000
if last_three < 2 or not is_prime(last_three):
return False
idx_digit_sum = sum(int(d) for d in str(index))
if idx_digit_sum < 2 or not is_prime(idx_digit_sum):
return False
return True
def mystery(x):
# Convert to integer
if isinstance(x, str):
if not x.isdigit():
return False
n = int(x)
else:
n = int(x)
# Condition: n > 10^18
if n <= 10**18:
return False
# Check that n+1 is a power of two
m = n + 1
if m <= 0 or (m & (m - 1)) != 0:
return False
# Get exponent p
p = m.bit_length() - 1
# Check that p is prime (small)
if p < 2:
return False
i = 2
while i * i <= p:
if p % i == 0:
return False
i += 1
# Now check that n is indeed 2^p - 1
if (1 << p) != m:
return False
# Lucas-Lehmer test for Mersenne prime 2^p - 1
s = 4
M = n # which should be 2^p - 1
for _ in range(p - 2):
s = (s * s - 2) % M
if s != 0:
return False
# Check sum of digits of n is prime
digit_sum = sum(int(d) for d in str(n))
if digit_sum < 2:
return False
j = 2
while j * j <= digit_sum:
if digit_sum % j == 0:
return False
j += 1
# Check number of digits is prime
dl = len(str(n))
if dl < 2:
return False
k = 2
while k * k <= dl:
if dl % k == 0:
return False
k += 1
return True