Bounded Gaps
Bounded-gap results proved that primes come within some fixed finite distance infinitely often. That is a major structural theorem about prime clustering, but it is still not the same as proving infinitely many twin primes.
Bounded gaps is one of the most important phrases in the modern twin-prime story, but it is also one of the easiest to blur into something it does not say. This page keeps the claim precise and connects it back to the broader conjecture and gap story.
A bounded-gap theorem says there is some fixed number B so that infinitely many prime pairs differ by at most B. The key point is that the same finite bound works infinitely often. This is already much stronger than merely observing many small gaps in computation, because it is a theorem about endless recurrence rather than a pattern in one large checked range.
Before these results, it was not known whether primes could be proved to recur within any fixed finite distance infinitely many times. The breakthrough turned that possibility into a theorem and changed how mathematicians talk about small prime gaps. It moved the conversation from heuristic expectation to theorem-level progress on a nearby version of the twin-prime problem.
A bounded-gap theorem does not identify one exact gap and say that it repeats forever. It proves that at least one finite gap inside a bounded range repeats infinitely many times. That is a strong structural claim, but it leaves open which specific gaps are doing the work. The distinction between some bounded gap and the exact gap 2 is the center of the whole interpretation problem.
Twin primes require the exact gap of 2. Bounded-gap theorems only prove that some finite bound works. Even if that bound is much smaller than earlier ones, it is still not the same as isolating the exact twin-prime pattern. This is why people say bounded gaps is very close to the twin-prime conjecture without saying it solves it.
From a distance, 'primes come very close together infinitely often' sounds almost identical to 'twin primes happen infinitely often.' The difference only becomes clear when you focus on the word exact. Twin primes ask for one exact gap. Bounded-gap theorems prove repeated small proximity without pinning the answer down to 2. That one missing step is exactly where the conjecture still remains open.
These theorems prove that local prime clustering is a real structural phenomenon, not just something suggested by computations or heuristics. That is why bounded gaps sits so close to the center of the modern twin-prime story. It shows that primes do keep returning near one another in a provable way, even if the final gap-2 statement is still beyond reach.
This page works best as a precision tool. Use it when headlines or summaries make progress sound closer to a completed proof than it really is. Then move back to the twin-prime conjecture page, the Zhang page, Theory, or the prime-gaps page to place the bounded-gap statement inside the larger research story. Analysis can then help you compare the explanatory idea with real spacing patterns in a chosen finite range.
Use these links to keep reading or jump back into the live number views.
Use the Zhang page when you want the breakthrough framed around the person and the 2013 result.
The Current Progress tab keeps the wider bounded-gap picture connected to the conjecture itself.
The conjecture page keeps the exact gap-2 question separate from nearby theorem progress.
Analysis lets you compare the article idea with gap patterns in a concrete finite range.
The prime-gaps page gives the larger spacing framework around the bounded-gap breakthrough.