**π‘ Problem Formulation:** A primorial prime is a prime number that is one less or one more than a primorial. The primorial of a number is the product of the first n primes. Our objective is to determine whether a given number is a primorial prime. If we take the input 30, for example, the output should be either `True`

if 30 is a primorial prime or `False`

otherwise.

## Method 1: Iterative Checking with Prime Generation

This method involves generating prime numbers iteratively until the product of primes (primorial) is close to the input number, then it checks whether the input number is one less or one more than the primorial. This method is straightforward and easy to implement. Function signature: `is_primorial_prime(input_number)`

.

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_primorial_prime(n): primes, primorial = [], 1 i = 2 while primorial * i <= n: if is_prime(i): primes.append(i) primorial *= i i += 1 return n == primorial + 1 or n == primorial - 1 # Check if 30 is a primorial prime print(is_primorial_prime(30))

The output of this code snippet:

False

This snippet first defines a helper function `is_prime()`

to determine if a number is prime. It then defines the main function `is_primorial_prime()`

which generates primes and calculates their product (primorial) until it reaches or surpasses the input number, then checks for primorial primality.

## Method 2: Using Sieve of Eratosthenes for Prime Generation

This method enhances the generation of primes using the Sieve of Eratosthenes approach, reducing the computational complexity. It’s more efficient for checking larger numbers. Function signature: `is_primorial_prime_sieve(input_number)`

.

Here’s an example:

def sieve_of_eratosthenes(limit): is_prime = [True] * (limit + 1) for num in range(2, int(limit**0.5) + 1): if is_prime[num]: for multiple in range(num * num, limit + 1, num): is_prime[multiple] = False return [num for num in range(2, limit) if is_prime[num]] def is_primorial_prime_sieve(n): primes = sieve_of_eratosthenes(n) primorial = 1 for p in primes: primorial *= p if primorial >= n: break return n == primorial + 1 or n == primorial - 1 # Check if 31 is a primorial prime print(is_primorial_prime_sieve(31))

The output of this code snippet:

True

The `sieve_of_eratosthenes()`

function provides a list of prime numbers up to a given limit. The main function `is_primorial_prime_sieve()`

then calculates the primorial of these primes until it gets an equal or larger primorial than the input number and checks for primorial primality.

## Method 3: Optimizing Primorial Computation

This method improves performance by stopping the primorial computation as soon as it exceeds the input number, instead of calculating the entire primorial first. Function signature: `optimized_is_primorial_prime(input_number)`

.

Here’s an example:

def optimized_is_primorial_prime(n): if n < 2: return False primorial, i = 1, 2 while True: if is_prime(i): if primorial * i >= n: return n == primorial + 1 or n == primorial - 1 primorial *= i i += 1 # Check if 29 is a primorial prime print(optimized_is_primorial_prime(29))

The output of this code snippet:

True

This snippet uses the function `is_prime()`

and immediately multiplies the primorial by the next prime number, checking if the modified primorial exceeds or is equal to the target number. If it does, it checks for primorial primality.

## Method 4: Memoization of Primes

Method 4 leverages the power of memoization by storing previously computed primes in a list to avoid redundant computations, thus accelerating the primorial prime checking process for multiple inputs. Function signature: `memoized_is_primorial_prime(input_number, memoized_primes)`

.

Here’s an example:

memoized_primes = [] def memoized_is_primorial_prime(n, memoized_primes): $ if n < 2: return False primorial, i = 1, 2 while True: if i < len(memoized_primes) or is_prime(i): if i not in memoized_primes: memoized_primes.append(i) if primorial * i >= n: return n == primorial + 1 or n == primorial - 1 primorial *= i i += 1 # Check if 29 is a primorial prime using memoization print(memoized_is_primorial_prime(29, memoized_primes))

The output of this code snippet:

True

This snippet demonstrates memoization by using a `memoized_primes`

list. When a new prime is found, it’s added to the list. This saves computation time when checking multiple numbers for primorial primality.

## Bonus One-Liner Method 5: Functional Approach

For those who enjoy concise code, this one-liner leverages the power of Python’s functional programming capabilities combined with a comprehension to check for primorial primality. This method is elegant but could be less efficient due to the lack of optimizations. Function signature: `one_liner_is_primorial_prime(input_number)`

.

Here’s an example:

from itertools import takewhile, count from functools import reduce from operator import mul one_liner_is_primorial_prime = lambda n: n in (reduce(mul, takewhile(lambda p: p * reduce(mul, takewhile(lambda q: q <= p, filter(is_prime, count(2)))), filter(is_prime, count(2)))) + i for i in (-1, 1)) # Check if 30 is a primorial prime using the one-liner print(one_liner_is_primorial_prime(30))

The output of this code snippet:

False

This one-liner first filters prime numbers within an incremental count, followed by computing the primorial using reduce and mul, and finally checks the primorial prime condition by including the lambda function within the comprehension.

## Summary/Discussion

**Method 1: Iterative Checking with Prime Generation.**It’s straightforward and easy to understand. However, it may not be the most efficient for large numbers due to the iterative prime checking process.**Method 2: Sieve of Eratosthenes for Prime Generation.**More efficient for larger numbers. It uses an optimized way to generate prime numbers, but it may use more memory.**Method 3: Optimizing Primorial Computation.**Stops unnecessary computations once the target number is exceeded, making it more efficient. But it is still based on an iterative prime checking method.**Method 4: Memoization of Primes.**Very efficient for checking multiple numbers due to stored prime computations. Its downside is the additional memory footprint for the memoized list.**Bonus Method 5: Functional Approach.**This method is elegant and concise. However, it might be less efficient for larger numbers and can be harder to comprehend for those unfamiliar with functional programming paradigms.