**Preliminaries**

**Sets A and B:**

- A={6k−1∣k∈Z}A = (Primes congruent to -1 modulo 6)
- B={6k+1∣k∈Z}B = (Primes congruent to +1 modulo 6)
*All primes greater than 3 fall into either set A or set B.*

- Twin primes are pairs of prime numbers (p, p + 2) differing by 2.

- The number of primes less than or equal to n, denoted by π(n), is asymptotically equivalent to n/ln(n) as n approaches infinity.

**Dirichlet’s Theorem on Arithmetic Progressions:**

- For any coprime integers a and d, the arithmetic progression a + nd contains infinitely many primes.

- There are infinitely many prime numbers.

- Provides an upper bound on the number of primes in an arithmetic progression: π(x; q, a) ≤ (2 + o(1)) * (x / (φ(q) * ln(x))) where π(x; q, a) counts primes less than or equal to x within the progression, and φ(q) is Euler’s totient function.

- Any number that is:
- An element of either set A or set B,
- And not a product of two elements from sets A or B (e.g., AA, AB, or BB), … must be a prime number.

- If p is a prime number in set A and p + 2 is composite, then the difference between p and any prime factor of p + 2 is strictly greater than 2. This holds true even when considering the combined contributions of all the prime factors of p + 2.

**Key Properties**

**Prime Representation:**

- All prime numbers greater than 3 can be expressed in either the form 6k – 1 (set A) or 6k + 1 (set B).

**Prime Factors:**

- If a number in set A or set B is composite, its prime factors must also belong to set A or set B.

**Symmetry:**

- Sets A and B are symmetrical around zero.

**The Argument**

**Assumption:**

*Assume, for the sake of contradiction, that there are only finitely many twin primes.*

**Consequence:**If true, there exists a largest twin prime pair (P, P + 2). This would imply that for any prime p > P, p + 2 cannot be prime.**Contradiction****of****Infinite Primes in A and B:**Dirichlet’s theorem ensures that both sets A and B contain infinitely many primes. This means we can always find a prime number p in set A that is greater than P + 2.

**Exploring p + 2:**

- Since p ∈ A, p + 2 must belong to set B. We have two cases:

(1) *Case 1: p + 2 is prime.*

- This immediately forms a twin prime pair with p, contradicting our assumption that (P, P + 2) is the largest twin prime pair.

(2) *Case 2: p + 2 is composite.*

- Since p + 2 is composite and in set B, it must be divisible by a product of two or more elements from sets A and B.
- The Gap Lemma ensures that any prime factor q of p + 2 that is in set B must be at least 4 units away from p. Therefore, it is impossible for p + 2 to be formed by multiplying p with a prime number that is only 2 units away. This contradiction highlights the impossibility of p + 2 being composite under our initial assumption.

**Contradiction with Dirichlet’s Theorem:**

- This means that for any prime number p greater than P + 2 within set A, the number p + 2 cannot be prime.
- This would imply that there are no twin primes beyond a certain point in the arithmetic progression 6k – 1 (set A). However, this directly contradicts Dirichlet’s Theorem, which guarantees an infinite number of primes within this progression.

**Density of Twin Primes**

**Decreasing Density:**

- The PNT tells us that the density of primes decreases as numbers grow larger. This means twin primes become less frequent as we look at larger numbers.

**Non-zero Density:**

- We can use the Brun-Titchmarsh Theorem to establish an upper bound on the density of twin primes. The theorem shows that while twin primes become less frequent, they never completely disappear.

**Zhang’s Result:**

- Yitang Zhang proved that there are infinitely many prime pairs with a bounded gap (less than 70 million). This result directly contradicts the assumption that there are only finitely many twin primes, providing strong evidence for the infinitude of twin primes.

**Conclusion**

Our assumption that there are finitely many twin primes has led to a contradiction with established theorems and properties of primes. The infinite nature of primes in sets A and B, the non-zero density of twin primes, and Zhang’s result on bounded gaps all point to the conclusion that there must be infinitely many twin primes.