Comprehensive Argument for the Infinitude of Twin Primes


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:

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

Prime Number Theorem (PNT):

  • 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.

Euclid’s Theorem:

  • There are infinitely many prime numbers.

Brun-Titchmarsh Theorem:

  • 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.

Semiotic Prime Theorem:

  • 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.

Gap Lemma:

  • 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.


  • Sets A and B are symmetrical around zero.

The Argument


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:


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.