Twin Prime Conjecture
No. The twin prime conjecture remains unsolved. Modern progress shows that primes come within some bounded distance infinitely often, but that is not the same as proving infinitely many gap-2 pairs.
If you only need the answer in one line, it is no. This page exists to make the next sentence clear too: important progress has happened, but the exact twin-prime claim still has not been proved, and that last distinction is where many summaries become misleading.
The twin prime conjecture claims that there are infinitely many pairs of prime numbers that differ by exactly 2. It is not a claim about small gaps in general. It is a claim about the exact gap of 2 repeating forever, no matter how far out on the number line you go.
News about bounded gaps between primes can sound very close to the twin prime conjecture. If you hear that primes come close together infinitely often, it is easy to assume that twin primes were proved too. That last step is exactly the part that remains open. The misunderstanding usually comes from replacing the word exact with the word close, even though those two ideas are not interchangeable in a theorem statement.
Modern theorems show that there exists some fixed finite bound B such that infinitely many prime pairs differ by at most B. That is a major breakthrough, because it proves recurring small gaps. But the bound is not known to be 2. A useful comparison is this: proving that some gap inside a bounded range keeps returning is already a theorem, while proving that the particular gap 2 returns infinitely often is still open.
This is one of the clearest examples in number theory of progress versus proof. The field has moved much closer to the conjecture, but the exact statement mathematicians want to prove is still unresolved. Finite examples, large computations, and bounded-gap theorems all matter, but none of them by themselves complete the final argument for infinitely many twin primes.
If you want the longer explanation of the open problem itself, move next to Twin Prime Conjecture Explained. If you want the best-known modern breakthrough nearby, go to the Zhang page. If you want to see why the distinction between finite evidence and proof matters visually, open the Lab or Analysis and compare what a real finite range can show with what a theorem about infinity would need to establish.
Use these links to keep reading or jump back into the live number views.
Use the longer conjecture explainer when you want the exact statement, what infinity means here, and what remains unproved.
Use the Zhang page when you want the cleanest explanation of what bounded gaps actually proved.
Theory collects the larger progress picture in one place.
The Lab helps connect the short answer back to visible gap-2 examples without confusing finite evidence with proof.