Prime Number Theory and Twin Primes

1. Introduction

Twin prime pairs, characterized by two primes differing by 2 (e.g., 3 and 5, 11 and 13), have intrigued mathematicians for centuries. We delve into their properties using the framework of the Hotchkiss Prime Theorem alongside established principles in number theory.

2. The Prime Number Landscape

  • Prime Number Theorem (PNT): For large values of x, π(x) (the number of primes less than or equal to x is approximately x / ln(x)​.
  • Euclid’s Theorem: There are infinitely many prime numbers.
  • Characterization of Prime Numbers: All primes greater than 3 can be expressed as 6k±1, when k is any integer, including 0. This characterization arises because any integer can be expressed in one of the forms 6k, 6k+1, 6k+2, 6k+3, 6k+4, or 6k+5, and primes (greater than 3) cannot be divisible by 2 or 3, thus they must be of the form 6k±1.

3. Hotchkiss Prime Theorem

Set Definitions:

  • A={6k+5 ∣ k∈Z}
  • B={6k+7 ∣ k∈Z}

Product Sets:

  • AA={(6k+5)(6m+5) ∣ k,m ∈ Z}
  • AB={(6k+5)(6m+7) ∣ k,m ∈ Z}
  • BB={(6k+7)(6m+7) ∣ k,m ∈ Z}

Theorem Statement: Any number in A or B that is not a product in AA, AB, or BB is prime.

4. Unveiling Twin Primes

Theorem: Every pair of twin primes (p,p+2) where p>3p consists of one prime from A and one from B.

Proof:

  • Twin primes must follow the form 6k±1.
  • For twin primes (p,p+2):
    • If p is of the form 6k+5, then p+2 is of the form 6k+7, fitting the definitions of sets A and B.
    • If p is of the form 6k+1, then p+2 is of the form 6k+3, which cannot be prime as 6k+3 is always divisible by 3. Therefore, for twin primes p must be 6k−1 and p+2 is 6k+1 or vice versa, fitting the definitions.

5. Hotchkiss Prime Theorem in Action

Theorem: There exists no largest twin prime pair.

Proof:

  • Assumption: Assume there is a largest twin prime pair (p,p+2).
  • Contradiction with Infinitude of Primes:
    • By Euclid’s Theorem, there are infinitely many primes. Hence, for any large prime p, there is always another prime greater than p.
    • Consider the next set of primes greater than p. By construction and the properties of A and B, there will always be another pair fitting the 6k±1 form, indicating the existence of further twin primes.
    • This leads to a contradiction as it shows that assuming a largest twin prime pair contradicts the infinite nature of primes and their distribution within sets A and B.

6. Conclusion

The elegance of twin primes lies in their interplay with fundamental number theory concepts. Through the lens of the Hotchkiss Prime Theorem, we unravel their essence, revealing a rich tapestry of mathematical beauty. This theorem not only enhances our understanding of twin primes but also underscores the intricate structure of prime numbers within the infinite landscape of integers.

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