Proposition · Number Theory · Open Since 1849
There exist infinitely many prime numbers p such that p + 2 is also prime.
∃ infinitely many p ∈ ℙ : p + 2 ∈ ℙ
No proof exists as of 2026. In 2013, Yitang Zhang proved infinitely many prime pairs exist within a gap of 70 million — the first finite bound ever established. James Maynard and the Polymath Project reduced this to 246. The target is a gap of exactly 2.
Definitions
Prime numbers — integers divisible only by 1 and themselves — are the atoms of arithmetic. Among them, twin primes appear as pairs separated by exactly two, with a single even number between them.
The conjecture, open since Alphonse de Polignac stated it formally in 1849, asks whether these pairs continue forever. We have found hundreds of millions of them. We cannot prove they never stop.
Definition I
Prime Number
A natural number greater than 1 with no positive divisors other than 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
Definition II
Twin Prime
A pair of primes (p, p+2) separated by exactly two.
(3,5) · (11,13) · (17,19) · (29,31) · (41,43)…
Progress
Every entry below represents the proven upper bound on prime gaps at a given moment in history. The target is 2. Watch the reduction — and appreciate how far remains.
The Target
2
A proven prime gap of exactly 2 would confirm the Twin Prime Conjecture. The current record stands at 246. The distance from 246 to 2 is not arithmetic — it may require mathematics that does not yet exist.
Twin Prime of the Day
—
"We have found hundreds of millions of twin prime pairs. We have no proof they ever stop. We have no proof they never stop."
No proof exists. No deadline for one.
Exploration
Drag the slider to explore integers up to any limit. Twin primes are highlighted in slate blue. Notice how they thin out — but never seem to disappear entirely.
Computational Frontier
Finding large twin primes requires distributed computing projects running for months. These are among the largest numbers ever proven prime by human computation.
| Pair — form (p, p+2) | Digits | Year | Discoverer |
|---|---|---|---|
|
2996863034895 · 21290000 − 1 2996863034895 · 21290000 + 1 Record |
388,342 | 2016 | PrimeGrid |
|
3756801695685 · 2666669 − 1 3756801695685 · 2666669 + 1 |
200,700 | 2011 | PrimeGrid |
|
65516468355 · 2333333 − 1 65516468355 · 2333333 + 1 |
100,355 | 2009 | PrimeGrid |
|
2003663613 · 2195000 − 1 2003663613 · 2195000 + 1 |
58,711 | 2007 | Vautier, McKibbon et al. |
|
100314512544015 · 2171960 − 1 100314512544015 · 2171960 + 1 |
51,780 | 2006 | PrimeGrid |
Each entry shows both primes in the pair. The −1 and +1 confirm a gap of exactly 2. Digit count applies to each prime individually. Writing them out in full would require more paper than exists.