Prime Numbers and Multiples of 24:


Prime numbers exhibit fascinating properties when examined through algebraic identities and divisibility rules. One intriguing fact is that for any prime number , the expression is always divisible by 24. This article explains why this statement is true through number theory and proofs.

1. Understanding the Statement

If is a prime number greater than or equal to 5, then:


In mathematical terms, this means:


For example:

• If .
• If .
• If .

In each case, the result is divisible by 24.

2. Why Is This True? (Algebraic Proof)

We can factor the expression :


For any prime :

• is an odd number because the only even prime is 2.
• , , and are three consecutive integers.

Since these are consecutive numbers:

• One of these numbers is divisible by 2 (in fact, two of them are because two out of any three consecutive numbers are even).
• One of these numbers is divisible by 3 because every set of three consecutive integers includes a multiple of 3.

Thus, the product is divisible by both and 3, meaning:


3. Modulo Argument: Another Way to See It

When is an odd prime:

• must be either or modulo 6:
  
  
• If .
• If .

In both cases:


4. Python Code to Verify the Property

Here’s a Python script to check this property for any prime number:

def is_prime(n):
    if n <= 1:
        return False
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False
    return True

def check_property(p):
    if p < 5:
        print(f"{p} does not satisfy the condition (p >= 5 required).")
        return False
    return (p**2 - 1) % 24 == 0

for num in range(5, 101):
    if is_prime(num) and check_property(num):
        print(f"{num}^2 - 1 is a multiple of 24.")

5. Why Does This Only Apply to ?

The only primes less than 5 are and :

• If (not divisible by 24).
• If (not divisible by 24).

Thus, the theorem holds only for .

6. Deeper Mathematical Insight

• The structure of three consecutive numbers ensures the presence of factors of 8 and 3.
• This property is related to quadratic residues in modular arithmetic.
• The property generalizes to other patterns involving prime powers and modular arithmetic.

7. Fun Facts About This Property

• The result follows from the structure of odd numbers and their neighbors.
• Similar properties hold for higher powers of primes and their differences.
• This property appears in problems involving congruences and prime decomposition.
• It is an example of how simple arithmetic can reveal deeper truths in number theory.

8. Conclusion

For any prime number , the expression is always a multiple of 24 due to the inherent divisibility of consecutive numbers. This beautiful fact connects prime numbers, modular arithmetic, and basic algebra.

References

• AoPS: Prime Numbers
• Wikipedia: Prime Numbers
• Hardy, G. H. An Introduction to the Theory of Numbers (2008).
• Kummer, E. Number Theory: An Introduction to Modular Forms (1996).
• Kline, M. Mathematical Thought from Ancient to Modern Times (1972).