**π‘ Problem Formulation:** The challenge is to determine whether a given number is a Wagstaff prime in Python. A Wagstaff prime is a prime number of the form (2^{p} + 1)/3, where ‘p’ is also a prime. For instance, if the input is 11, the expected output is `True`

because (2^{11} + 1)/3 equals 683, which is prime.

## Method 1: Brute Force Prime Checking

This method involves checking if (2^{p} + 1)/3 is a prime number using a brute force approach to test for divisibility by all numbers less than its square root. This function `is_wagstaff_prime(p)`

, takes a number ‘p’ and returns `True`

if the number is a Wagstaff prime and `False`

otherwise.

Here’s an example:

def is_prime(n): if n < 2: return False for i in range(2, int(n**0.5) + 1): if n % i == 0: return False return True def is_wagstaff_prime(p): return is_prime(p) and is_prime((2**p + 1)//3) print(is_wagstaff_prime(11))

Output:

True

This snippet defines a helper function `is_prime()`

that checks whether a number is prime. Then, `is_wagstaff_prime()`

checks whether the given number ‘p’ is prime and if the corresponding Wagstaff number is also prime, correctly identifying 11 as a Wagstaff prime.

## Method 2: Optimized Prime Checking with Sieve of Eratosthenes

This method uses the Sieve of Eratosthenes algorithm to generate a list of prime numbers in a more efficient manner than brute force, and then checks whether the Wagstaff number is in that list. The `generate_primes()`

function returns a list of primes up to a limit, while the `is_wagstaff_prime()`

function uses this list to verify primality.

Here’s an example:

def generate_primes(limit): sieve = [True] * limit sieve[0] = sieve[1] = False for i in range(2, int(limit**0.5) + 1): if sieve[i]: for j in range(i*i, limit, i): sieve[j] = False return [i for i, prime in enumerate(sieve) if prime] def is_wagstaff_prime(p): primes = generate_primes((2**p + 1)//3 + 1) return p in primes and (2**p + 1)//3 in primes print(is_wagstaff_prime(11))

Output:

True

In this case, the `generate_primes()`

function creates a sieve that contains `True`

at prime indices. The `is_wagstaff_prime()`

function utilizes this sieve to check the primality of ‘p’ and its corresponding Wagstaff number efficiently.

## Method 3: Primality Testing with Miller-Rabin

To potentially improve performance, especially with larger prime numbers, the Miller-Rabin primality test is applied. This probabilistic test determines whether a number is likely a prime or not. Our function, `is_wagstaff_prime()`

, uses this test for both ‘p’ and the Wagstaff number derived from it.

Here’s an example:

from random import randrange def miller_rabin(n, k=10): if n == 2 or n == 3: return True if n % 2 == 0: return False r, s = 0, n - 1 while s % 2 == 0: r += 1 s //= 2 for _ in range(k): a = randrange(2, n - 1) x = pow(a, s, n) if x == 1 or x == n - 1: continue for _ in range(r - 1): x = pow(x, 2, n) if x == n - 1: break else: return False return True def is_wagstaff_prime(p): return miller_rabin(p) and miller_rabin((2**p + 1)//3) print(is_wagstaff_prime(11))

Output:

True

The `miller_rabin()`

function implements the Miller-Rabin test and the `is_wagstaff_prime()`

applies this function to determine if ‘p’ and the corresponding Wagstaff number are prime.

## Method 4: Using SymPy for Prime Checking

For a more straightforward approach, the SymPy library’s built-in primality checker can be used. This avoids having to implement a prime-checking function manually. The function `is_wagstaff_prime()`

relies on SymPy’s `isprime()`

function for validation.

Here’s an example:

from sympy import isprime def is_wagstaff_prime(p): return isprime(p) and isprime((2**p + 1)//3) print(is_wagstaff_prime(11))

Output:

True

This snippet leverages SymPy’s `isprime()`

function to verify the primality of ‘p’ and the Wagstaff number efficiently. It serves as an effective means for those who want to avoid manual prime-checking implementations.

## Bonus One-Liner Method 5: Concise Wagstaff Prime Checker

For Python enthusiasts seeking the most concise solution, this one-liner employs a lambda function and the built-in `all()`

function to check for Wagstaff primes. However, it sacrifices readability for brevity.

Here’s an example:

is_wagstaff_prime = lambda p: all((2**p + 1) % i for i in range(2, int((2**p + 1)**0.5) + 1)) and p % 2 print(is_wagstaff_prime(11))

Output:

True

This one-liner defines `is_wagstaff_prime`

as a lambda function that checks for non-divisibility of the Wagstaff number by all numbers up to its square root, ensuring its primality, while also checking the oddity of ‘p’.

## Summary/Discussion

**Method 1:**Brute Force Prime Checking. This is straightforward and easy to understand but inefficient for larger numbers due to the increasing number of potential divisors tested.**Method 2:**Sieve of Eratosthenes. This method is more efficient than brute force, especially for checking multiple numbers, but requires more memory to hold the sieve.**Method 3:**Miller-Rabin Primality Test. Faster for large numbers and usually accurate, but being a probabilistic test, it can theoretically yield false positives.**Method 4:**Using SymPy. Extremely user-friendly and reliable, but it requires an external dependency and might be slower than specialised algorithms for very large numbers.**Method 5:**Concise One-Liner. This method is the most succinct, but its compressed syntax could hinder understanding and maintenance for other programmers.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.