The twin prime conjecture asks whether there are infinitely many pairs of primes that differ by 2 — twin primes. It is one of the oldest open problems in number theory and the site of some of the most dramatic recent progress.
The conjecture
Twin primes are pairs of primes differing by 2: , , , , , , and so on.
The conjecture: there are infinitely many such pairs.
History
The conjecture is implicit in work going back at least to Euclid. It has been formally studied since at least the 19th century, when mathematicians began asking general questions about gaps between primes.
Computationally, twin primes have been found very far out. The largest known twin primes as of 2024 have approximately 388,000 decimal digits.
The breakthrough
For a long time, no one could even prove that there are infinitely many prime pairs within any fixed distance of each other. Then in 2013, the Chinese-American mathematician Yitang Zhang — a relatively unknown lecturer at the University of New Hampshire — published a stunning result:
There are infinitely many prime pairs with .
The key word is infinitely many. No one had proved an analogous statement for any finite bound before.
Within months, a collaborative project led partly by Terence Tao — the Polymath8 project — reduced the bound from 70 million to around 246. Under strong conjectural assumptions (the Elliott-Halberstam conjecture), the bound drops to 12.
Getting all the way down to 2 — actual twin primes — remains open.
Why it matters
The twin prime conjecture is part of a larger circle of questions about additive structure among primes. Along with Goldbach’s conjecture, it sits at the intersection of multiplicative and additive number theory — a historically fertile but notoriously difficult intersection.
Zhang’s breakthrough and the Polymath follow-up transformed the field. They suggested that real progress on twin primes was possible, and they showcased the power of collaborative mathematics in the internet age. Polymath8 was largely conducted through blog comments and shared documents, open for anyone qualified to contribute.
As of 2025, the bounded-gap result is a theorem; the full twin prime conjecture is still a conjecture. But the gap, in both senses, is smaller than it has ever been.
Frequently asked
What counts as a twin prime?
Two primes that differ by exactly 2. The first few: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43). The conjecture is that infinitely many such pairs exist.
What is the Zhang-Tao breakthrough?
In 2013 Yitang Zhang proved that there are infinitely many prime pairs differing by at most 70,000,000. The bound was rapidly reduced through the Polymath8 collaboration (led partly by Terence Tao) to 246 and, assuming some conjectures, to 12. Getting to 2 — actual twin primes — is the remaining challenge.