Twin Prime Question
No one has proved it yet, but many mathematicians expect the answer is yes. That mix of strong expectation and missing proof is exactly why the twin prime conjecture matters.
If you ask whether there are infinitely many twin primes, the careful answer is: this is strongly expected, but still unproved. This page is written for the direct search-style version of the question, so it separates the short answer, the mathematical expectation, and the theorem-level situation without assuming you already know the formal conjecture language.
Mathematicians have not proved that infinitely many twin-prime pairs exist. So the fully honest answer is: it is unproved. At the same time, the dominant expectation is that twin primes do continue forever. This means the best short answer is not simply yes or no. It is unproved, but strongly expected to be true.
The expectation comes from heuristics, large-scale computation, and the way twin-prime patterns keep appearing across many finite ranges. For example, pairs such as (11, 13), (17, 19), and (29, 31) show the basic pattern early, and much larger examples continue to appear as computation pushes farther out. None of this is a proof, but it is a major reason the conjecture remains plausible.
Some nearby results are proved. Bounded-gap theorems show that primes come within some fixed finite distance infinitely often. Brun's theorem proves an important fact about the sparsity of twin primes if they continue. But no theorem currently proves that the exact gap 2 occurs infinitely many times. That final step from small bounded gaps to the specific twin-prime pattern is still missing.
The solved-or-not page is built for the direct status question: was the conjecture proved? This page is answering a slightly different search-style question: what should I believe about infinitely many twin primes? The answer there is more nuanced. The conjecture is unsolved, but the expectation remains yes. Keeping those two page types separate helps avoid mixing up proof status with mathematical belief.
The formal conjecture page states the exact problem and explains what 'infinitely many' means with more precision. This page is the more conversational companion. It is meant to be the landing page for a reader who asks the obvious question first and only afterward wants the stricter theorem-versus-conjecture framing.
If you want the formal mathematical statement, move next to Twin Prime Conjecture Explained. If you want the direct solved-or-not clarification, use the short-answer page. If you want to see the visible finite pattern before returning to the theory, open the Lab or Explorer and look at real twin-prime examples in a selected range.
Use these links to keep reading or jump back into the live number views.
Use the full conjecture explainer for the exact statement, what infinity means here, and what remains unproved.
Use the short-answer clarification when you want the proof-status question handled as directly as possible.
The twin-primes page gives the best plain-language introduction to the pattern itself before you think about the infinite question.
Theory keeps the conjecture, research approaches, and current progress in one place.