The theorem in plain language
Zhang proved that prime numbers do not drift apart forever. Instead, there exists at least one fixed distance B so that infinitely many prime pairs appear with a gap no larger than B.
What this page clarifies
Zhang's breakthrough is often mentioned in one sentence and misunderstood in the next. This page separates the theorem itself, why it mattered so much, what later work changed, and why bounded gaps still does not mean twin primes were proved.
The theorem in notation and in English
If p_n denotes the nth prime number, Zhang's result can be summarized by saying that the liminf of p_(n+1) - p_n is finite. In plain language, that means there is some fixed bound that keeps recurring between consecutive primes infinitely often. The theorem does not tell us that the recurring bound is 2, only that it does not have to grow without limit.
Why this mattered immediately
Before Zhang, no one had proved that primes come close together infinitely often in any bounded way. His result changed the field by turning a long-standing expectation into a theorem.
Why the breakthrough was so surprising
Mathematicians already had strong heuristic reasons to believe in small prime gaps, but a proof required new control over how primes are distributed in arithmetic progressions. Zhang supplied enough of that control to push bounded gaps from a hope into a theorem. That is why the result was treated as a genuine field-changing breakthrough rather than just one more improved estimate.
What it did not prove
Zhang did not prove that the gap is 2 infinitely often. The twin prime conjecture asks for the exact gap of 2, while bounded-gap results only guarantee that some finite gap occurs infinitely many times.
What happened after Zhang
Zhang opened the door, and the next wave of work pushed through it quickly. The Polymath Project reduced the original numerical bound through large-scale collaboration, while James Maynard and Terence Tao developed methods that broadened bounded-gap theory beyond Zhang's first theorem. Modern discussion of prime gaps now includes all of these advances, but Zhang's 2013 result remains the turning point.
Why the result still belongs in the twin-prime story
The theorem showed that prime clustering is not just a heuristic guess. It established that small prime gaps recur across infinitely many scales, which is one of the strongest pieces of progress connected to the twin prime problem.
How to see the idea in TwinPrimeExplorer
The site cannot reproduce the proof, but it can help you build intuition for what bounded gaps are about. In Analysis, the Gaps tab lets you inspect repeated small spacings in a selected range. In the Lab, you can watch clusters of twin-prime candidates and centers appear rather than treating prime gaps as an abstract theorem statement. That shift from theorem language to visible structure is exactly why this page belongs next to the tools.
Where to go next
Use these links to keep reading or jump back into the live number views.
See the broader progress picture
Theory puts Zhang, Polymath, and Maynard-Tao into the same timeline.
Need the short answer first?
Use the short answer page for a clear explanation of why bounded gaps still do not solve the twin prime conjecture.
Look at gap structure directly
Analysis helps connect the idea of small gaps to concrete ranges and repeated spacing.
Step back to prime gaps
Use the prime-gaps page when you want bounded gaps placed inside the broader spacing story first.
Check the bounded-gaps definition
The Glossary keeps the distinction between bounded gaps and twin primes short and clear.