def mystery(x):
    if not isinstance(x, int) or x <= 1:
        return False
    
    sqrt_x = int(x ** 0.5)
    if sqrt_x * sqrt_x != x:
        return False
    
    # Check if sqrt_x is prime
    if sqrt_x < 2:
        return False
    for i in range(2, int(sqrt_x ** 0.5) + 1):
        if sqrt_x % i == 0:
            return False
    return True

def mystery(x):
    if not isinstance(x, int) or x <= 0:
        return False
    if (x * x + 1) % 2 != 0:
        return False
    y_sq = (x * x + 1) // 2
    y = int(y_sq ** 0.5 + 0.5)
    return y * y == y_sq

def mystery(x):
    if not isinstance(x, int) or x <= 0:
        return False
    
    # Check if x is a perfect square
    sqrt_x = int(x ** 0.5 + 0.5)
    if sqrt_x * sqrt_x != x:
        return False
    
    # Check if x is also a triangular number
    # Triangular numbers: T_n = n(n+1)/2
    # We need: 1 + 8x = (2n+1)²
    disc = 1 + 8 * x
    sqrt_disc = int(disc ** 0.5 + 0.5)
    if sqrt_disc * sqrt_disc != disc:
        return False
    # Additional check: sqrt_disc must be odd
    return sqrt_disc % 2 == 1

def mystery(x):
    if not isinstance(x, int) or x <= 0:
        return False
    
    if (x * x - 4) % 5 != 0:
        return False
    
    y_sq = (x * x - 4) // 5
    if y_sq < 0:
        return False
    
    y = int(y_sq ** 0.5 + 0.5)
    return y * y == y_sq

def mystery(x):
    if not isinstance(x, int) or x <= 0:
        return False
    
    y_sq = 3 * (x * x - 1)
    y = int(y_sq ** 0.5 + 0.5)
    return y * y == y_sq