There are infinitely many pairs of primes that differ by exactly 2.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The twin prime conjecture, that there are infinitely many primes p for which p + 2 is also prime, is one of the oldest unsolved problems in number theory. It is stated in the affirmative by Hardy and Littlewood in 1923 and is the case of gap 2 of de Polignac's 1849 conjecture, but no proof exists, and no serious mathematician claims one: the purported proof by Arenstorf in 2004 was withdrawn after a gap was found, and the stream of later preprints announcing proofs has not been accepted by the field.
The evidence that the conjecture is true is nonetheless strong by the standards mathematicians apply to open problems. Heuristically, modelling the primes as a random set with the right local corrections predicts that the number of twin prime pairs below x is asymptotically 2 C₂ x/(log x)², and exhaustive counts (more than 8 × 10^14 pairs below 10^18) match that prediction to eight significant figures. Rigorously, the conjecture has been proved up to a bounded relaxation in each of its two parameters: Chen showed in 1973 that infinitely many primes p have p + 2 prime or a product of two primes, and the work of Zhang, Maynard and Tao (2013), sharpened by the Polymath project, showed that infinitely many prime pairs differ by at most a fixed constant, in fact by at most 246. Under the generalised Elliott–Halberstam conjecture the bound drops to 6, but the parity obstruction in sieve theory is understood to prevent these methods from reaching 2 without a genuinely new idea.
Nothing points the other way. Brun's 1919 theorem that the sum of the reciprocals of the twin primes converges shows the twin primes are sparse but is consistent with their infinitude. Expert opinion is essentially unanimous that the conjecture is true; what remains open is only its proof, and a proof of the Hardy–Littlewood asymptotic or of a level of distribution beyond the parity barrier would settle it.
Full reasoning: the evidence and decisions behind this verdict
Status. The mathematics guidance assigns "supported" to an open claim with evidence mathematicians count and "unsupported" to a conjecture resting on plausibility alone. The twin prime conjecture is firmly in the first category: it has a quantitative heuristic that is confirmed numerically at every scale computed, and two independent families of proven weakenings. This also matches how the graph assesses Goldbach's conjecture, a sibling problem with the same evidential profile.
Evidence for. (1) The Hardy–Littlewood heuristic argument: the twin prime count is asymptotically 2 C₂ x/(log x)² with C₂ ≈ 0.66016. The count of twin prime pairs below 10^18, 808,675,888,577,436 (Oliveira e Silva's computation), agrees with the prediction to a relative error of order 10^-8, and sieve upper bounds (Brun; Bombieri–Davenport; the best constant is due to Wu) show the true count is at most about 3.4 times the prediction, so the heuristic is not merely fitting data but is bracketed by theorem from above. This is the main reason experts hold the conjecture as near-certain. (2) The proven approximations: Chen's theorem, that infinitely many primes p have p + 2 prime or a product of two primes, and the bounded-gaps theorems, infinitely many prime pairs differ by at most a fixed constant and by at most 246 (Zhang, Annals of Mathematics 179 (2014); Maynard, Annals 181 (2015); Polymath, Research in the Mathematical Sciences 1 (2014)). These show the pattern the conjecture asserts persists under either relaxation.
Evidence against. None credible. The parity problem of sieve theory explains why the current methods stall at "prime or product of two primes" and at gap 246 rather than 2; it is an obstruction to proof techniques, not evidence that the statement is false. Brun's theorem (convergence of the reciprocal sum) constrains density and is neutral on infinitude.
Instances. The Quanta Magazine article on Zhang (www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/) states the conjecture as an open problem and takes no side; its stance was corrected from affirms to poses on reading it whole. Wolfram MathWorld (mathworld.wolfram.com/TwinPrimeConjecture.html, updated September 2026) records that the conjecture is unproven, that Arenstorf's 2004 proof was retracted, that bounded gaps do not resolve it, and judges it "almost certain to be true", quoting Hardy and Wright and Shanks to the same effect. A web search on the state of the problem in 2025–2026 turned up several self-announced proofs (a conference-proceedings paper, Zenodo and Cambridge Open Engage preprints, institutional repository entries) and no acceptance of any of them by the number-theory community; MathWorld's September 2026 entry continues to list the problem as unsolved. No source denies the conjecture.
Credence. 0.97 that the claim is true as stated: the heuristic and its numerical confirmation are strong, the proven approximations rule out the most natural ways it could fail, and it has withstood a century of attention without a counterexample or a competing heuristic. The residual reflects the fact that heuristics for primes have occasionally misled (Skewes-type phenomena in the distribution of primes, Maier's theorem on short intervals) and that no lower bound of the right order is known. Verdict confidence 0.88: the only defensible alternative status is "unsupported" on a strict reading that only proof counts as evidence, and that reading is contrary to the mathematics guidance and to how the graph treats comparable conjectures.
What would change the verdict. A refereed proof (or a machine-checked proof of the reviewed formal statement, which asserts infinitude of the set of primes p with p + 2 prime) would move the status to verified. A disproof is not expected but would move it to contradicted; a proof of a Polignac-type or Hardy–Littlewood-type statement would also settle it.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Two families of theorems establish weakenings of the conjecture in each of its two directions. Relaxing the gap, infinitely many prime pairs differ by at most a fixed constant, and specifically infinitely many prime pairs differ by at most 246; relaxing primality instead, infinitely many primes p have p + 2 prime or a product of two primes. Given that the conjecture is the common sharpening of both (gap exactly 2, both members prime), these results show that the behaviour it asserts is real up to a bounded gap or one extra prime factor, which raises confidence in the conjecture without proving it.
The premises are established theorems and are not in doubt: Chen's result that infinitely many primes p have p + 2 prime or a product of two primes and the bounded-gaps theorems that infinitely many prime pairs differ by at most a fixed constant, indeed by at most 246. What they yield is corroboration rather than implication: neither a bounded gap nor an almost-prime companion entails a gap of exactly 2 with both members prime, and the parity obstruction in sieve theory is precisely why these methods stop short of the conjecture. The argument therefore raises confidence in the conjecture by showing its asserted pattern survives every relaxation that has been proved, and it can do no more than that.
Treating the primes as a random set of density 1/log n and correcting for the divisibility constraints that the pair (n, n + 2) must jointly satisfy yields the prediction that the number of twin prime pairs below x is asymptotically 2 C₂ x/(log x)², a quantity that tends to infinity. Because the computed counts of twin primes (808,675,888,577,436 pairs below 10^18) agree with this prediction to within a relative error of order 10^-8, and sieve upper bounds confirm the count is at most a constant multiple of the prediction, the asymptotic is credible, and it implies that there are infinitely many twin primes.
The inference is valid: if the twin prime count is asymptotically 2 C₂ x/(log x)², the count is unbounded and there are infinitely many twin primes. The argument's whole weight rests on that asymptotic, which is itself an open conjecture, so the argument delivers not proof but the strong probabilistic evidence its premise carries: a heuristic bracketed from above by proven sieve bounds and matched by exhaustive counts to eight significant figures. The caveat is that agreement of a heuristic with computed data, however close, is not a proof, and heuristics about primes have occasionally failed at scales beyond computation.
Provenance
Where this claim has been said, linked to its canonical form.
the twin primes conjecture, which proposes that there are infinitely many pairs of primes that differ by only 2
Quanta Magazine feature on Yitang Zhang's bounded-gaps theorem, introducing the twin primes conjecture as one of mathematics' oldest unsolved problems that Zhang's result approaches without settling.
The source's own evidence bears what it asserts. A science-journalism account of Zhang's 2013 bounded-gaps theorem. It presents the twin primes conjecture as an unsolved problem, correctly reports that Zhang proved only that some gap below 70 million occurs infinitely often, and takes no position on whether the conjecture itself is true.
The first version states that there are infinitely many pairs of twin primes (Guy 1994, p. 19). It is not known if there are infinitely many such primes (Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to be true.
Reference entry on the twin prime conjecture. It states that the conjecture is unproven, records the retracted Arenstorf 2004 proof and the bounded-gaps results of Zhang and Polymath as not resolving it, and in its own voice judges the conjecture almost certainly true, citing Hardy and Wright's and Shanks's assessments of the evidence. Recorded as a hedged affirmation rather than a flat one.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.