# 5 Best Ways to Check if a Number is a Wagstaff Prime in Python

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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 (2p + 1)/3, where ‘p’ is also a prime. For instance, if the input is 11, the expected output is `True` because (211 + 1)/3 equals 683, which is prime.

## Method 1: Brute Force Prime Checking

This method involves checking if (2p + 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.