Check whether N is a Dihedral Prime Number or not in Python

A Dihedral Prime Number is a number that remains prime when viewed on a 7-segment display in normal orientation and when rotated 180 degrees (upside down). The number must contain only digits that look the same when rotated: 0, 1, 2, 5, and 8.

For example, if we have n = 1181, it appears as 1881 when rotated upside down (since 2 becomes 5 and 5 becomes 2), and both numbers are prime.

Algorithm Steps

To check if a number is a dihedral prime ?

1. Check if all digits are valid (0, 1, 2, 5, 8 only)
2. Create the upside-down version by swapping 2?5
3. Check if the original number is prime
4. Check if the upside-down version is prime
5. Check if the reverse is prime
6. Check if the reverse of upside-down is prime

Implementation

def is_prime(n):
    """Check if a number is prime"""
    if n < 2:
        return False
    if n == 2:
        return True
    if n % 2 == 0:
        return False
    
    for i in range(3, int(n**0.5) + 1, 2):
        if n % i == 0:
            return False
    return True

def reverse(n):
    """Reverse the digits of a number"""
    return int(str(n)[::-1])

def up_side_down(n):
    """Create upside-down version by swapping 2<->5"""
    temp = n
    total = 0
    
    while temp > 0:
        digit = temp % 10
        if digit == 2:
            digit = 5
        elif digit == 5:
            digit = 2
        total = total * 10 + digit
        temp //= 10
    
    return total

def is_dihedral_prime(n):
    """Check if n is a dihedral prime number"""
    # Check if all digits are valid for 7-segment display rotation
    temp = n
    while temp > 0:
        digit = temp % 10
        if digit in [3, 4, 6, 7, 9]:
            return False
        temp //= 10
    
    # Check if all four variations are prime
    upside_down_n = up_side_down(n)
    reverse_n = reverse(n)
    reverse_upside_down_n = reverse(upside_down_n)
    
    return (is_prime(n) and 
            is_prime(upside_down_n) and 
            is_prime(reverse_n) and 
            is_prime(reverse_upside_down_n))

# Test the function
n = 1181
result = is_dihedral_prime(n)
print(f"Is {n} a dihedral prime? {result}")

# Test with a few more examples
test_cases = [11, 181, 1181]
for num in test_cases:
    print(f"{num}: {is_dihedral_prime(num)}")

Is 1181 a dihedral prime? True
11: True
181: True
1181: True

How It Works

The algorithm first validates that the number contains only digits that remain recognizable when rotated 180° on a 7-segment display. These are: 0, 1, 2, 5, and 8. Digits 3, 4, 6, 7, and 9 would not be readable upside down.

Next, it creates the upside-down version by swapping digits 2 and 5, since these appear as each other when rotated. Finally, it checks that all four variations (original, upside-down, reverse, and reverse upside-down) are prime numbers.

Key Points

• Only digits 0, 1, 2, 5, 8 are valid for dihedral primes
• Digits 2 and 5 swap when rotated 180°
• All four number variations must be prime
• The algorithm handles both rotation and reflection

Conclusion

A dihedral prime must contain only 7-segment compatible digits and remain prime in all orientations. This creates a very restrictive set of numbers that are both mathematically interesting and visually symmetric.