Check if N is a Factorial Prime in Python

A factorial prime is a prime number that is either one less than or one more than a factorial of any positive integer. For example, 719 is a factorial prime because 719 = 6! - 1 = 720 - 1, where 6! = 720.

In this article, we'll learn how to check if a given number N is a factorial prime in Python.

What is a Factorial Prime?

A factorial prime has the form n! ± 1, where:

• n! + 1 (one more than factorial)
• n! - 1 (one less than factorial)

The number must also be prime to qualify as a factorial prime.

Algorithm

To check if a number is a factorial prime, we follow these steps:

• First, check if the number is prime
• If not prime, return False
• Calculate factorials incrementally
• Check if the number equals factorial ± 1
• Return True if found, otherwise False

Implementation

Here's the complete solution to check if N is a factorial prime:

from math import sqrt

def isPrime(num):
    if num <= 1:
        return False
    if num <= 3:
        return True
    if num % 2 == 0 or num % 3 == 0:
        return False
    for i in range(5, int(sqrt(num)) + 1, 6):
        if num % i == 0 or num % (i + 2) == 0:
            return False
    return True

def isFactorialPrime(num):
    if not isPrime(num):
        return False
    
    factorial = 1
    i = 1
    
    while factorial <= num + 1:
        factorial *= i
        if num + 1 == factorial or num - 1 == factorial:
            return True
        i += 1
    
    return False

# Test with example
num = 719
result = isFactorialPrime(num)
print(f"{num} is a factorial prime: {result}")

# Test with more examples
test_numbers = [2, 3, 5, 23, 719]
for n in test_numbers:
    print(f"{n}: {isFactorialPrime(n)}")

719 is a factorial prime: True
2: True
3: True
5: True
23: True
719: True

How It Works

The algorithm works in two phases:

1. Prime Check: Uses an optimized primality test that checks divisibility up to ?n
2. Factorial Check: Calculates factorials incrementally and checks if the number equals n! ± 1

Example Verification

Let's verify why 719 is a factorial prime:

import math

# Check if 719 is prime
def verify_factorial_prime():
    num = 719
    
    # Calculate factorials to find the match
    for i in range(1, 10):
        factorial = math.factorial(i)
        if num == factorial - 1:
            print(f"{num} = {i}! - 1 = {factorial} - 1")
            return True
        elif num == factorial + 1:
            print(f"{num} = {i}! + 1 = {factorial} + 1")
            return True
    
    return False

verify_factorial_prime()

719 = 6! - 1 = 720 - 1

Time Complexity

The time complexity is O(?n + k), where:

• O(?n) for the primality test
• O(k) for factorial calculation, where k is the number of factorials to check

Conclusion

A factorial prime is a prime number that equals n! ± 1 for some positive integer n. Our solution efficiently checks both primality and the factorial condition to determine if a number is a factorial prime.