💡 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! ± 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 SymPy’s 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.