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.