Conjectured in the general form by A. de Polignac, 1849 · quantitative form by Hardy & Littlewood, 1923 · field: analytic number theory · status: OPEN
How to read this page. We write p for a prime and ℕ, ℤ for the natural numbers and integers. A twin prime pair is a pair (p, p+2) with both members prime: (3,5), (5,7), (11,13), (17,19), (29,31), … We write pn for the n-th prime, so the prime gap is gn = pn+1 − pn, and π2(x) for the number of twin pairs with smaller member ≤ x. "log" is the natural logarithm. "liminf" is the smallest limit point of a sequence. Throughout, read the claim as a statement about an infinite set: the difficulty is entirely in the word "infinitely."
0 · The claim, in one line
Among the primes, the pattern (p, p+2) (two primes two apart) recurs forever. We can find billions of them by computer, but no one has proved they never run out.
∃∞ p : p prime ∧ (p+2) primeThere exist infinitely many primes p such that p + 2 is also prime.
1 · The object: what it is and why anyone cares
The primes are the multiplicative atoms of ℤ: every integer factors uniquely into them. Euclid proved around 300 BC that there are infinitely many primes. But how they are distributed is the central drama of number theory, and the governing fact is that primes thin out. The Prime Number Theorem (Hadamard and de la Vallée Poussin, independently, 1896) makes this precise: the density of primes near x is about 1 / log x, so the average gap between consecutive primes near x grows without bound.
π(x) ~ x / log x, average gap near x ≈ log x → ∞Primes become ever rarer; typical spacing grows like log x.
Against that backdrop of ever-widening average gaps, the twin prime conjecture asks whether the smallest possible gap between odd primes (a gap of exactly 2) keeps occurring forever. (A gap of 1 is impossible beyond (2,3), since one of any two consecutive integers is even.) So the question is really: does the minimum keep happening even as the mean runs off to infinity?
Why chase this particular question?
It is the simplest unanswered question about additive structure in the primes. Primes are defined multiplicatively, but twins are an additive constraint (p+2). The clash between the multiplicative definition and additive patterns is the deep theme that also drives Goldbach's conjecture and the Hardy–Littlewood k-tuple conjectures. Twin primes is the smallest, cleanest case.
It is a test of how "random" the primes are. Heuristically the primes behave like a random set of density 1 / log x. A naïve independence guess predicts infinitely many twins; the real mathematics is in turning a heuristic into a theorem, and in correcting the heuristic for the arithmetic bias that primes are never even.
Its difficulty mapped a fundamental limit of a whole technology. The repeated failure to prove it exposed the parity problem of sieve theory (Section 5): arguably the single most important structural obstruction discovered in 20th-century analytic number theory.
2 · The precise statement and the mathematical setup
The conjecture in its plainest form:
#{ p ≤ x : p, p+2 both prime } → ∞ as x → ∞The count of twin pairs is unbounded: equivalently, the set of twin primes is infinite.
The quantitative (Hardy–Littlewood) form
In 1923, in Partitio Numerorum III, G. H. Hardy and J. E. Littlewood went much further than mere infinitude. Their first conjecture predicts the exact asymptotic density of twins:
π2(x) ~ 2C2 ∫2x dt / (log t)2 ~ 2C2 · x / (log x)2The number of twin pairs up to x grows like a constant times x/(log x)², the "square" of the prime density times a correction factor.
Here C2 is the twin prime constant, an explicit product over odd primes:
C2 = ∏p ≥ 3 p(p−2)/(p−1)2 ≈ 0.6601618158…The arithmetic correction to the "primes are independent" guess; the factor 2 and this product encode the local densities modulo every prime.
The factor 1/(log x)2 is exactly what you'd guess if the events "p prime" and "p+2 prime" were independent (each probability ~1/log x). The constant 2C2 is the honest correction: it accounts for the fact that these events are not independent: once p is odd (forced, for p > 2), p+2 is automatically odd too, which boosts the chance both are prime, while constraints modulo 3, 5, 7, … pull the other way. This conjecture matches computation to extraordinary precision but is completely unproven: far beyond what we can establish. Proving the mere infinitude (Section 0) is already open; proving the asymptotic is strictly harder.
The admissible-tuple framework
The modern language that unifies twins with all related problems: a finite set ℋ = {h1, …, hk} of distinct integers is admissible if for every prime p the residues h1, …, hk (mod p) do not occupy all p residue classes. If they did, then for every n one of n+hi would be divisible by p, killing all hope of simultaneous primality. The pair ℋ = {0, 2} is admissible (modulo 2 both residues are 0, so the class 1 is missed; modulo any p ≥ 3 two residues cannot fill all p classes). The twin conjecture is then: the admissible tuple {0, 2} yields infinitely many n with both n and n+2 prime.
Twin pairs are the tightest possible clustering of odd primes. As you walk right along the number line they grow sparse: but do they ever stop? That is the open question.
3 · Every connection: what it touches and what it unlocks
Brun's theorem and the Brun constant
The first deep theorem about twins is a convergence result. In 1919 Viggo Brun proved (by inventing the modern combinatorial sieve, now called the Brun sieve) that the sum of reciprocals of the twin primes converges:
B2 = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ⋯ = 1.902160583…Brun's constant: unlike the primes themselves (whose reciprocals diverge), the twins are sparse enough that their reciprocal sum is finite.
This is a striking contrast with Euler's theorem that ∑ 1/p = ∞. It tells us the twins are genuinely rare: but, crucially, convergence does not decide finiteness. A convergent sum can have infinitely many terms (e.g. ∑ 1/n2). So Brun's theorem, for all its importance, is consistent with both answers to the conjecture. It gives an upper bound on the twin count: π2(x) = O(x/(log x)2), the correct order of magnitude: but no lower bound.
The Hardy–Littlewood k-tuple conjecture
Twins are the first case of a vast web: the Hardy–Littlewood prime k-tuples conjecture asserts that every admissible tuple ℋ is realized infinitely often (all entries simultaneously prime), with a predictable density. This single conjecture would imply the twin prime conjecture, the infinitude of prime triplets, cousin primes (p, p+4), sexy primes (p, p+6), and de Polignac's full 1849 conjecture that every even number is a prime gap infinitely often.
What it is NOT equivalent to
The Riemann Hypothesis does not settle it. RH (and even the Generalized RH) controls the error term in prime counting, but twins concern a binary correlation that RH does not reach. One can assume RH freely and still not prove twins. This surprises newcomers and is worth stating flatly: RH ⇏ twin primes.
The Elliott–Halberstam conjecture (EH) gets close but not there. EH asserts primes are extremely evenly distributed across arithmetic progressions (level of distribution θ = 1; Bombieri–Vinogradov unconditionally gives θ = 1/2). Assuming EH, Maynard proved there are infinitely many prime pairs differing by at most 12; assuming a generalized EH, Polymath8 reduced this to 6. Neither reaches 2: see Section 5.
4 · Every serious attack: and exactly how far each got
Sieve theory (1919 →): bounding and approximating
Viggo Brun, 1919. Brun sieve ⟹ B2 < ∞ and π2(x) ≪ x/(log x)2. Correct upper order of magnitude; no lower bound on infinitude.
Chen Jingrun, announced 1966, full proof 1973. The high-water mark of the "approximation" strategy: there are infinitely many primes p such that p+2 is either prime or a product of two primes (a semiprime, P2). Written p + 2 = P2. This is agonizingly close ("prime or almost-prime") yet the last step from "P2" to "prime" is exactly the step sieves cannot take (Section 5).
Small-gaps program (2005 →): the breakthrough decade
Goldston, Pintz, Yıldırım (GPY), 2005–2009. Introduced a new weighted sieve and proved liminfn (pn+1 − pn)/log pn = 0: primes come infinitely often much closer than average. They showed that any improvement of Bombieri–Vinogradov past θ = 1/2 would give bounded gaps outright. They could not supply that improvement; the problem stalled here for years.
Yitang Zhang, 2013 (published Annals of Mathematics, 2014). The seismic result. Zhang proved the first finite bound on infinitely-recurring gaps, with no unproven hypothesis:
liminfn→∞ (pn+1 − pn) < 70 000 000Zhang 2013: some even gap below 70 million occurs between consecutive primes infinitely often.
Zhang's engine was a hard-won extension of Bombieri–Vinogradov-type distribution to primes in progressions with smooth (friable) moduli, squeezing just past the θ = 1/2 barrier: enough to make GPY's machine fire.
Polymath8a, 2013. A massively collaborative online project optimized Zhang's argument, driving the bound from 70 000 000 down to 4680.
James Maynard, 2013 (published 2015); Terence Tao, independently, same period. A genuinely new multidimensional sieve that bypassed Zhang's hardest distribution input and was far more flexible. Maynard reached gap ≤ 600 unconditionally, and proved that for every m there are infinitely many intervals containing m+1 primes within bounded length.
Polymath8b, 2014. Combining Maynard–Tao with Polymath optimizations gave the current unconditional record:
liminfn→∞ (pn+1 − pn) ≤ 246The best proven bound as of today: infinitely often, two consecutive primes differ by at most 246.
Conditionally: assuming the Elliott–Halberstam conjecture the same methods give ≤ 12 (Maynard), and assuming a generalized form, ≤ 6 (Polymath8b). The gap between 246 and the target 2 is not a matter of more optimization: it is a wall of principle.
Analytic / additive attacks
The Hardy–Littlewood circle method. The circle method triumphed on "ternary" additive problems (Vinogradov, 1937: every large odd number is a sum of three primes). But twins is a binary problem, and for binary problems the "minor arcs" cannot be controlled: the method provably does not close (same obstruction that leaves binary Goldbach open).
5 · Why it breaks down: the honest crux
Three distinct walls block a proof. The first is the deepest and is the reason every sieve-based attack, no matter how clever, stops just short.
Wall 1: The parity problem of sieve theory (Selberg). Atle Selberg observed in the 1940s–50s that sieve methods are fundamentally blind to the parity of the number of prime factors. A sieve estimates how many integers in a set survive after removing multiples of small primes, but it cannot distinguish an integer with an even number of prime factors from one with an odd number. Formally, a "pure" sieve cannot tell the Liouville function λ(n) = (−1)Ω(n) from a constant. Primes have exactly one prime factor (odd parity); semiprimes have two (even parity). This is precisely the gap between Chen's theorem ("p+2 is prime or a semiprime") and the conjecture ("p+2 is prime"). The sieve delivers you to the doorstep of "P2" and then cannot, as a matter of structural principle, separate the primes from the two-prime-factor numbers hiding among them. Any proof must inject genuinely non-sieve arithmetic information: something that sees parity: and no such ingredient is known for this problem.
Wall 2: Bounded gaps are not a gap of 2, and the method provably cannot descend to 2. The Zhang–Maynard–Tao machine proves that some even gap ≤ 246 recurs infinitely often. It does not identify which gap, and in particular says nothing about the gap being exactly 2. The mechanism works with an admissible tuple ℋ of size k and shows that infinitely often at least two of {n+h1,…,n+hk} are simultaneously prime: but to force k = 2 (the tuple {0,2} itself, which is the twin conjecture) you would need a level of distribution θ far beyond anything provable, and even granting the strongest reasonable conjecture (generalized Elliott–Halberstam) the method caps out at gap 6. The number 6 is not an engineering limit: Polymath8b showed it is the exact barrier of this sieve, and the reason it cannot be pushed to 4 or 2 is, once again, the parity problem of Wall 1. The small-gaps revolution, for all its glory, is structurally incapable of reaching 2.
Wall 3: Binary additive correlations resist the circle method. Twin primes asks for a correlation: p prime AND p+2 prime. The Hardy–Littlewood circle method evaluates such counts via an integral of exponential sums over the "major arcs" (rational points with small denominator, giving the conjectured main term 2C2x/(log x)2) and "minor arcs" (the error). For ternary problems the minor-arc contribution is provably smaller than the main term; for a binary problem like twins (or binary Goldbach) there are not enough averaging variables, and no one can bound the minor arcs below the main term. The method computes the right prediction but cannot prove it is attained even once infinitely often. The predicted main term is a heuristic, not a lower bound.
A clearly-labelled FRINGE / speculative line: and why it has not closed the gap
Speculative (not a proof): random models and the Möbius/parity-breaking programme. Cramér's probabilistic model treats primes as independent random events of density 1/log x; refined (Hardy–Littlewood-corrected) versions predict the twin asymptotic with uncanny accuracy and match computation out to enormous ranges. This is powerful evidence but explicitly not a proof: a heuristic model can be "right" numerically while the true primes harbor a conspiracy that only shows up in the infinite limit; the model simply assumes away the correlations that constitute the entire difficulty. A more serious speculative programme aims to break parity by importing a "Type-I / Type-II" bilinear or dispersion estimate (in the spirit of the methods behind the ternary Goldbach and Chowla-correlation advances) that would see the Möbius/Liouville function non-trivially in this correlation. Progress has been made on averaged and almost-all versions of such correlations (e.g. results on autocorrelations of multiplicative functions), but nothing yet supplies the pointwise, parity-aware input that Wall 1 demands for the single tuple {0,2}. Fringe ideas occasionally surface: quantum/spectral analogies (primes as "energy levels"), p-adic or nonstandard-analysis reformulations, cellular-automaton pattern arguments: but to date none has produced a mechanism that distinguishes primes from semiprimes in the twin setting, which is the one thing that would actually matter.
6 · Status: confirmed
OPEN. As of 2026 the twin prime conjecture is unproven and unrefuted. What is rigorously known:
The twins are sparse: ∑ (1/p + 1/(p+2)) converges to Brun's constant B2 ≈ 1.902 (Brun 1919), and π2(x) ≪ x/(log x)2.
Infinitely often p+2 is prime or a semiprime (Chen 1973).
Infinitely often two consecutive primes differ by at most 246 (Zhang 2013; Maynard–Tao; Polymath8, 2014): conditionally ≤ 6 under generalized Elliott–Halberstam.
Computation has verified twins far beyond 1018; the largest known twin pair has hundreds of thousands of digits. None of this proves infinitude.
What a genuine proof would still require
A real proof of the conjecture must do at least one thing no current technique can: overcome the parity problem for the specific tuple {0,2}: i.e. supply arithmetic information that distinguishes primes (one prime factor) from numbers with two prime factors, pointwise and infinitely often. Equivalently, it would need to establish a lower bound of the correct order ≫ x/(log x)2 for the twin count, which no method (sieve, circle, or probabilistic) can presently reach even in principle. Proving the full Hardy–Littlewood asymptotic would be strictly harder still. A conditional proof under Elliott–Halberstam would not suffice, because even EH leaves the gap at 6, not 2. Barring a genuinely new idea that sees parity, the problem remains exactly where Brun, Selberg, and Zhang left it: close enough to touch, and walled off by principle.
Credit where it is due. Alphonse de Polignac (1849, general conjecture); G. H. Hardy and J. E. Littlewood (1923, quantitative form and the k-tuple conjecture); Viggo Brun (1919, the sieve and Brun's constant); Atle Selberg (the parity problem); Chen Jingrun (1966/1973, "prime + P2"); Daniel Goldston, János Pintz, Cem Yıldırım (2005–2009, GPY); Yitang Zhang (2013, the first bounded gap); the Polymath8 collaboration; and James Maynard and Terence Tao (2013–2015, the multidimensional sieve). This page reports their work; it proves nothing new. The unsolved problem is claimed by no one here: if it is ever solved, that proof is offered freely to the world.