**π‘ Problem Formulation:** Determining whether a number *n* is a dihedral prime is crucial in various mathematical and cryptographic applications. A dihedral prime is a prime number that remains prime when written in reverse. This article aims to provide Python-based methods to check for dihedral primes. For instance, given the input `n = 13`

, the output should indicate whether flipping to `31`

preserves its primality.

## Method 1: Traditional Prime Check With String Reversal

This method involves checking if *n* is prime and if the reverse of *n* is also prime. The `is_prime()`

function checks primality, and `str[::-1]`

reverses the string representation of *n*.

Here’s an example:

def is_prime(num): if num < 2: return False for i in range(2, int(num ** 0.5) + 1): if num % i == 0: return False return True def is_dihedral_prime(n): return is_prime(n) and is_prime(int(str(n)[::-1])) print(is_dihedral_prime(13))

Output:

`True`

This method defines a utility function to check for prime numbers and applies it to both the original number and its reversed form. If both are prime, *n* is a dihedral prime. This approach is straightforward and good for understanding the basics of dihedral primes.

## Method 2: Sieve of Eratosthenes with Post-Check

Improving efficiency, Method 2 uses the Sieve of Eratosthenes to generate a list of prime numbers up to a maximum limit and then checks if *n* and its reverse are in the list.

Here’s an example:

def sieve_of_eratosthenes(limit): prime = [True for i in range(limit+1)] p = 2 while (p * p <= limit): if (prime[p] == True): for i in range(p * p, limit+1, p): prime[i] = False p += 1 return prime def is_dihedral_prime(n, primes): return primes[n] and primes[int(str(n)[::-1])] primes = sieve_of_eratosthenes(100) print(is_dihedral_prime(31, primes))

Output:

`True`

Here, a list of boolean values indicates the primality for every number up to a specified limit. The method efficiently determines the primality status of *n* and its reverse without repeatedly checking for factors, which is particularly effective for checking multiple numbers.

## Method 3: Using a Prime Number Library

We can leverage Python libraries such as `sympy`

to perform the prime check, saving the time needed for algorithm implementation. The library’s prime check function is then applied to both *n* and its reverse.

Here’s an example:

from sympy import isprime def is_dihedral_prime(n): return isprime(n) and isprime(int(str(n)[::-1])) print(is_dihedral_prime(37))

Output:

`True`

This snippet uses SymPy’s built-in prime checking function, which employs optimized algorithms for prime testing. This method significantly reduces code complexity and is useful for quick checks and casual use. However, it requires installing an external library.

## Method 4: Recursive Prime Check with String Manipulation

Method 4 is a recursive implementation to check if *n* is prime in conjunction with checking if its reversed version is prime. Recursion breaks down the problem into smaller instances for easier calculation.

Here’s an example:

def is_prime_recursive(num, div=2): if num < 2: return False if div > int(num ** 0.5): return True if num % div == 0: return False return is_prime_recursive(num, div + 1) def is_dihedral_prime(n): return is_prime_recursive(n) and is_prime_recursive(int(str(n)[::-1])) print(is_dihedral_prime(73))

Output:

`True`

Recursively dividing *n* by numbers up to its square root, we verify its primality without iterative loops. This can be more intuitive for some, but recursive function calls may lead to higher memory usage on large input numbers.

## Bonus One-Liner Method 5: Lambda Expressions

A compact and functional approach to check for a dihedral prime is to use lambda expressions combined with a filter on range elements.

Here’s an example:

is_prime = lambda num: num > 1 and not any(num % i == 0 for i in range(2, int(num**0.5) + 1)) is_dihedral_prime = lambda n: is_prime(n) and is_prime(int(str(n)[::-1])) print(is_dihedral_prime(79))

Output:

`True`

The one-liner defines two lambda functions, one for checking prime numbers and another for dihedral primality using string reversal. This method is highly concise but may be less readable for some users, particularly those unfamiliar with functional programming paradigms.

## Summary/Discussion

**Method 1:**Traditional Prime Check With String Reversal. Strengths: Easy to understand; no external dependencies. Weaknesses: Not the most efficient for large numbers or repeated computations.**Method 2:**Sieve of Eratosthenes with Post-Check. Strengths: Efficient for multiple checks; good for upper-bounded problems. Weaknesses: Requires pre-computation and uses extra space.**Method 3:**Using a Prime Number Library. Strengths: Fast and easy to implement. Weaknesses: Depends on an external library.**Method 4:**Recursive Prime Check with String Manipulation. Strengths: Intuitive problem-solving approach. Weaknesses: Possible stack overflow for very large input numbers.**Method 5:**Lambda Expressions. Strengths: Very concise, functional approach. Weaknesses: Can be less readable and harder to debug.