def mystery(x):
if not isinstance(x, (str, int)):
return False
try:
n = int(x) if isinstance(x, str) else x
except:
return False
if n <= 0:
return False
s = str(n)
r = int(s[::-1])
if n == r:
return False
a = n + r
b = n - r
import math
return math.isqrt(a) ** 2 == a and round(abs(b) ** (1/3)) ** 3 == abs(b)
def mystery(x):
if not isinstance(x, int) or x <= 100:
return False
s = str(x)
if len(set(s)) != 1:
return False
def q(n):
return sum(map(int, str(n)))
def k(n):
r, d, v = [], 2, n
while d * d <= v:
while v % d == 0:
r.append(d)
v //= d
i = 1 # Obfuscation
d += 1
if v > 1: r.append(v)
return r
m = k(x)
return len(m) > 1 and q(x) == sum(map(q, m))
def mystery(x):
if not isinstance(x, (str, int)):
return False
try:
n = int(x) if isinstance(x, str) else x
except:
return False
if n <= 1:
return False
def p(v):
if v < 2:
return False
if v == 2:
return True
if v % 2 == 0:
return False
i = 3
while i * i <= v:
if v % i == 0:
return False
i += 2
return True
if not p(n):
return False
m = n + 1
if m & (m - 1):
return False
r = int(str(n)[::-1])
if r == n:
return False
if not p(r):
return False
d = sum(map(int, str(n)))
import math
if math.isqrt(d) ** 2 != d:
return False
k = m.bit_length() - 1
if not p(k):
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 1000:
return False
def _f(n):
res, temp, d = [], n, 2
while d * d <= temp:
if temp % d == 0:
res.append(d)
temp //= d
if temp % d == 0: return []
d += 1
return res + [temp] if temp > 1 else res
v = _f(x)
if len(v) < 3 or any((x - 1) % (p - 1) for p in v):
return False
s = str(x)
sd = sum(int(d) for d in s)
is_prime = sd > 1 and all(sd % i for i in range(2, int(sd**0.5) + 1))
if not is_prime:
return False
return '0' in s
def mystery(x):
# Convert to int if string
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int):
return False
if x <= 100000:
return False
# Primality test
def is_prime(n):
if n < 2:
return False
if n == 2:
return True
if n % 2 == 0:
return False
i = 3
while i * i <= n:
if n % i == 0:
return False
i += 2
return True
if not is_prime(x):
return False
# Fibonacci test: 5*x^2 ± 4 must be a perfect square
import math
s = 5 * x * x
a = s + 4
b = s - 4
if not (math.isqrt(a) ** 2 == a or math.isqrt(b) ** 2 == b):
return False
# Digit properties
s_str = str(x)
digits = [int(d) for d in s_str]
digit_sum = sum(digits)
if not is_prime(digit_sum):
return False
sum_squares = sum(d * d for d in digits)
if not is_prime(sum_squares):
return False
if x % 7 != 2:
return False
if x % 11 != 1:
return False
from functools import reduce
import operator
product = reduce(operator.mul, digits, 1)
if product <= 700:
return False
if len(s_str) % 2 != 0:
return False
return True
def mystery(x):
import math
try:
n = int(x)
if n <= 100: return False
except (ValueError, TypeError): return False
_q = lambda v: v > 1 and all(v % i for i in range(2, int(v**0.5) + 1))
s = str(n)
if '0' in s or not all(_q(int(s[i:])) for i in range(len(s))):
return False
c, history = n, {n}
while c != 1:
c = sum(int(d)**2 for d in str(c))
if c in history: return False
history.add(c)
if not _q(bin(n).count('1')):
return False
digit_sum = sum(int(d) for d in s)
if math.isqrt(digit_sum)**2 != digit_sum:
return False
p = 1
for d in s: p *= int(d)
return p % 10 == 6
def mystery(x):
# Convert to int if possible
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int) or x <= 100000:
return False
# Check that x is not prime
if x < 2:
return False
if x % 2 == 0:
if x == 2:
return False # 2 is prime, but x > 100000 so not encountered
# even and > 2 → composite
pass
else:
# odd number, check for divisors
i = 3
while i * i <= x:
if x % i == 0:
break
i += 2
else:
# no break → prime
return False
# Digit sum and product
s = str(x)
digit_sum = sum(int(ch) for ch in s)
if x % digit_sum != 0:
return False
digit_product = 1
for ch in s:
digit_product *= int(ch)
# Check product is a perfect cube
if digit_product == 0:
# 0 is a perfect cube (0³ = 0)
pass
else:
root = round(digit_product ** (1/3))
if root ** 3 != digit_product:
return False
# Taxicab check: at least two distinct representations as sum of two positive cubes
reps = []
max_val = int(x ** (1/3)) + 1
for a in range(1, max_val):
a3 = a ** 3
if a3 > x:
break
for b in range(a, max_val):
b3 = b ** 3
s3 = a3 + b3
if s3 > x:
break
if s3 == x:
reps.append((a, b))
if len(reps) >= 2:
break
if len(reps) >= 2:
break
return len(reps) >= 2
def mystery(x):
import math
try:
# Initial validation and conversion
n = int(x)
s = str(n)
nl = len(s)
if n <= 100:
return False
except (ValueError, TypeError):
return False
# Helper for primality testing
_q = lambda v: v > 1 and all(v % i for i in range(2, int(v**0.5) + 1))
# Condition 1: Circular Primality
# Every cyclic rotation of the digits must be prime
if not all(_q(int(s[i:] + s[:i])) for i in range(nl)):
return False
# Condition 2: Digital Sum Sequence (Keith Number Property)
# The number must appear in a sequence starting with its own digits,
# where each term is the sum of the previous 'nl' terms.
res = [int(d) for d in s]
while res[-1] < n:
res.append(sum(res[-nl:]))
# Condition 3: Digital Properties
# The sum of digits and the sum of the squares of digits must be prime.
d_list = [int(v) for v in s]
if res[-1] != n or not (_q(sum(d_list)) and _q(sum(v**2 for v in d_list))):
return False
# Condition 4: Divisor Count Property
# The number of divisors of (n - 1) must be a perfect square.
target, count = n - 1, 0
for i in range(1, int(target**0.5) + 1):
if target % i == 0:
count += 1 if i * i == target else 2
return math.isqrt(count)**2 == count
def mystery(x):
# Convert to int if possible
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int):
return False
if x <= 1000000:
return False
# Primality test
def is_prime(n):
if n < 2:
return False
if n == 2:
return True
if n % 2 == 0:
return False
i = 3
while i * i <= n:
if n % i == 0:
return False
i += 2
return True
if not is_prime(x):
return False
# Reverse and check
s = str(x)
r = int(s[::-1])
if r == x:
return False
if not is_prime(r):
return False
# Digit sum and product
digits = [int(d) for d in s]
digit_sum = sum(digits)
# Check digit sum is a perfect cube
cube_root = round(digit_sum ** (1/3))
if cube_root ** 3 != digit_sum:
return False
digit_product = 1
for d in digits:
digit_product *= d
if not is_prime(digit_product):
return False
# Modular conditions
if x % 7 != 6 or x % 11 != 6:
return False
# Binary ones count (popcount) must be a perfect square
import math
popcount = bin(x).count('1')
if math.isqrt(popcount) ** 2 != popcount:
return False
# Sum of squares of digits: half must be a perfect square
sum_squares = sum(d * d for d in digits)
if sum_squares % 2 != 0:
return False
half = sum_squares // 2
if math.isqrt(half) ** 2 != half:
return False
return True
def mystery(x):
import math
def _p(v):
if v < 2: return False
if v == 2: return True
if v % 2 == 0: return False
for i in range(3, int(v**0.5) + 1, 2):
if v % i == 0: return False
return True
try:
if isinstance(x, str):
n = int(x)
elif isinstance(x, int):
n = x
else:
return False
except (ValueError, TypeError):
return False
if n <= 1000:
return False
# Condition 1: Sophie Germain prime property
if not (_p(n) and _p(2 * n + 1)):
return False
# Condition 2: Digital properties
s = str(n)
if '0' not in s or s == s[::-1]:
return False
d = [int(c) for c in s]
# Digit sum and sum of squares must be prime
if not (_p(sum(d)) and _p(sum(i * i for i in d))):
return False
# Condition 3: Binary representation popcount must be a perfect square
pop = bin(n).count('1')
if round(pop**0.5)**2 != pop:
return False
# Condition 4: Modular properties
return n % 13 == 5 and n % 7 == 4