Twin prime conjecture

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The twin prime conjecture

   A twin prime is a pair of primes differing by $2$. Twin prime conjecture. There are infinitely many primes $q$ such that $q+2$ is prime. The conjecture is one of Landau's four problems; the source claims it is proved, but the supplied parser status gives no resolution evidence.

   source: Agostino Prástaro, “The Landau's problems. I-II”, arXiv:1208.2473 (2015).

2. The twin prime conjecture

   Let $p_n$ denote the $n$th prime and let $d_n=p_{n+1}-p_n$. The twin prime conjecture.

   $$\liminf\limits_{n \to \infty} d_n=2.$$

   This is the assertion that the gap $2$ occurs infinitely often between consecutive primes. It remains open.

   source: Janos Pintz, “Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture”, arXiv:1305.6289 (2013).

3. The twin prime conjecture

   A twin prime is a prime number that differs from another prime number by two. Twin prime conjecture. There are infinitely many pairs of twin primes. The conjecture is a famous open problem in number theory; although bounded gaps between infinitely many primes are known, no proof that infinitely many prime pairs differ by exactly two is currently known.

   source: Vineet Kumar, “Prime number generation and factor elimination”, arXiv:1411.3356 (2014).

4. The twin prime conjecture

   Twin prime conjecture. There are infinitely many primes $p$ such that $p+2$ is also a prime.

   This is one of the central open problems about gaps between consecutive primes. It is equivalent to the assertion that infinitely many integers $n$ make both $n$ and $n+2$ prime.

   source: Paweł Lewulis, “Almost primes in various settings”, arXiv:1806.09034 (2020).

5. The twin prime conjecture

   Let a twin prime be a pair of prime numbers differing by $2$, equivalently primes $p$ and $p+2$.

   Twin prime conjecture. There exist infinitely many twin primes.

   This is one of the classical open problems in number theory. The source notes that the conjecture remains open, while bounded-gap results show that infinitely many prime pairs occur with some fixed bounded difference; they do not establish the case of difference $2$.

   source: Mbakiso Fix Mothebe, “Sieve methods and the twin prime conjecture”, arXiv:1909.02205 (2023).

6. Twin Prime Conjecture

   Let $(p_n)_{n\in\mathbb{N}}$ denote the sequence of prime numbers in increasing order. Twin Prime Conjecture. There are infinitely many values $n\in\mathbb{N}$ such that

   $$p_{n+1}-p_n=2.$$

   This is the classical twin prime conjecture, asserting the existence of infinitely many pairs of primes differing by two. It remains unproved.

   source: Bryce Alan Christopherson, “Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems”, arXiv:2502.04311 (2025).

7. The twin prime conjecture

   Let $n$ range over the positive integers. Twin prime conjecture. There are infinitely many pairs of primes of the form $n,n+2$. This is the $k=2$ case of the prime-tuple prediction represented by Dickson's conjecture and remains open.

   source: Andrew Darlington, “Hopf-Galois structures on separable field extensions of degree related to Cunningham chains”, arXiv:2508.03384 (2025).

8. The twin primes conjecture

   Let $p$ and $p+2$ be a pair of twin primes, meaning that both numbers are prime. Twin primes conjecture. There are infinitely many pairs of twin primes $p,p+2$. The conjecture is one of the central open problems in analytic number theory. Bounded-gap results show that infinitely many pairs of primes occur within a fixed distance, but it remains unknown whether infinitely many pairs differ by exactly $2$.

   source: David Lowry-Duda, “A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture”, arXiv:1401.7555 (2014).

9. Euclid's conjecture on infinitely many twin primes

   Let $p$ range over prime numbers. Euclid's conjecture. There exist infinitely many prime pairs $p$ and $p+2$, such as $3,5$; $5,7$; $11,13$; $17,19$; and so on. This is the twin prime conjecture, a classical open problem asserting that the gap $2$ occurs infinitely often between consecutive primes.

   source: Guangchang Dong, “Infinitely many pairs of primes p and p+2”, arXiv:1406.4996 (2019).