**π‘ Problem Formulation:** In this article, we aim to determine whether a given integer `n`

is a factorial prime. A factorial prime is a prime number that is one less or more than a factorial (n! Β± 1). For instance, given the input `n = 23`

, the desire output is true since 23 is a prime and equals 4! – 1.

## Method 1: Brute Force Factorial Prime Check

This method sequentially calculates factorials and checks if the result plus or minus one is a prime number. It is easy to implement but computationally expensive for large numbers. The function specification involves iterating through natural numbers until the factorial exceeds `n`

and checking for prime status.

Here’s an example:

import math def is_prime(num): if num < 2: return False for i in range(2, int(math.sqrt(num)) + 1): if num % i == 0: return False return True def is_factorial_prime(n): fact = 1 i = 2 while fact <= n: fact *= i if fact - 1 == n or fact + 1 == n: return is_prime(n) i += 1 return False print(is_factorial_prime(23))

Output: `True`

The code snippet defines two functions: `is_prime()`

which checks if a number is prime, and `is_factorial_prime()`

which looks for a factorial within one of `n`

that is also a prime. The example confirms that 23 is indeed a factorial prime.

## Method 2: Optimized Factorial Calculation

Instead of computing the full factorial for each step, this method keeps track of the previous factorial computation and simply multiplies it by the next integer. This approach reduces computational overhead. It also employs an optimized prime checking function.

Here’s an example:

def is_prime_optimized(num): if num <= 1: return False if num <= 3: return True if num % 2 == 0 or num % 3 == 0: return False i = 5 while i * i <= num: if num % i == 0 or num % (i + 2) == 0: return False i += 6 return True def is_factorial_prime_optimized(n): fact = 1 i = 2 while True: fact *= i if fact > n: break if fact - 1 == n or fact + 1 == n: return is_prime_optimized(n) i += 1 return False print(is_factorial_prime_optimized(23))

Output: `True`

This snippet improves performance by using an optimized prime checking algorithm in `is_prime_optimized()`

and a loop that breaks once the factorial exceeds `n`

to avoid unnecessary calculations in `is_factorial_prime_optimized()`

.

## Method 3: Prime Sieve Precomputation

This method uses a prime sieve to precompute the primes up to a limit and then checks if `n`

is one more or one less than any calculated factorial. It is faster for multiple queries but requires an upfront calculation of the prime sieve.

Here’s an example:

def prime_sieve(limit): sieve = [True] * (limit + 1) sieve[0], sieve[1] = False, False for i in range(2, int(limit**0.5) + 1): if sieve[i]: for j in range(i*i, limit + 1, i): sieve[j] = False return [x for x in range(2, limit+1) if sieve[x]] def is_factorial_prime_sieve(n, primes): fact = 1 for i in range(2, n+1): fact *= i if fact > n: break return n - 1 in primes or n + 1 in primes primes = prime_sieve(100) # Should be at least n to be reliable print(is_factorial_prime_sieve(23, primes))

Output: `True`

This approach uses a sieve algorithm in `prime_sieve()`

to generate a list of prime numbers, and then checks if `n`

is a factorial prime in `is_factorial_prime_sieve()`

. This method is particularly useful if we want to check multiple numbers against a known list of primes.

## Method 4: Utilizing the SymPy Library

This method leverages the external SymPy library for prime checking and factorial calculations. While this introduces a dependency, it simplifies the code and improves readability. The library is known for being robust and efficient for symbolic mathematics.

Here’s an example:

from sympy import factorial, isprime def is_factorial_prime_sympy(n): i = 2 fact = factorial(i) while fact <= n: if fact - 1 == n or fact + 1 == n: return isprime(n) i += 1 fact = factorial(i) return False print(is_factorial_prime_sympy(23))

Output: `True`

By using SymPyβs built-in `factorial()`

and `isprime()`

functions, this example minimizes the amount of custom code needed, thus making it more maintainable and easier to understand.

## Bonus One-Liner Method 5: Functional Approach

For Python enthusiasts looking for elegance and conciseness, a functional programming style one-liner is used. This approach is not recommended for large calculations but is a neat, compact representation of the solution.

Here’s an example:

from sympy import isprime from math import factorial is_factorial_prime_one_liner = lambda n: isprime(n) and any(n == factorial(x) - 1 or n == factorial(x) + 1 for x in range(n)) print(is_factorial_prime_one_liner(23))

Output: `True`

This one-liner uses a lambda function in combination with Python’s `any()`

function and list comprehension. It checks if `n`

is prime and if it is equal to any factorial – 1 or factorial + 1 for all values up to `n`

.

## Summary/Discussion

**Method 1:**Brute Force Factorial Prime Check. Straightforward but not efficient for large numbers. Good for simplicity and basic understanding of the problem.**Method 2:**Optimized Factorial Calculation. Slightly better in performance due to an optimized prime check. Helps to understand optimization techniques.**Method 3:**Prime Sieve Precomputation. Efficient for multiple checks against a large set of numbers. It illustrates an important algorithmic concept in prime number theory.**Method 4:**Utilizing the SymPy Library. Leverages a powerful library for a clean and concise solution. Great when using Python for rapid development and prototyping.**Bonus Method 5:**Functional Approach. One-liner that is elegant but not the best for performance. Showcases Python’s abilities for concise code.

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.