Twin prime conjecture

A prime is a whole number greater than one divisible only by one and itself. Prime pairs such as 11 and 13 differ by two, as do 17 and 19. Do pairs with exactly this gap keep appearing forever, beyond every fixed point on the number line?[1] [2]

A prime is a whole number greater than one divisible only by one and itself. Do primes separated by exactly two occur beyond every bound? The conjecture says that for every $B$, some prime $p>B$ also has $p+2$ prime. After the pair $3,5$, any such pair must have the form $6m-1,6m+1$: otherwise one member would be divisible by two or three. This necessary pattern is not sufficient. It gives primes $11,13$ at $m=2$, but at $m=6$ it gives $35,37$, where thirty-five is composite.

List the primes in increasing order as $p_1,p_2,\ldots$. Twin primes would make the limiting smallest repeatedly occurring consecutive gap equal to two, written $\liminf_{j\to\infty}(p_{j+1}-p_j)=2$. A 2014 Polymath theorem established that infinitely many consecutive prime gaps are at most 246. That is a dated unconditional milestone, not a claim about the current best bound. It ensures repeated bounded gaps without requiring any of them to be two.[2]

Theorems guarantee infinitely many pairs of primes with gaps at most a fixed constant larger than two. That gives a bounded distance, but does not force the distance to equal two. The exact two-step pattern remains the target.[2]

Bounded-gap theorems leave the two-gap assertion open. The July 2026 function-field result concerns a different arithmetic setting.[1] [2]

Knowing how many primes occur in a long stretch does not tell us whether any sit two steps apart. Twin primes test how these multiplicative building blocks line up locally. An answer would explain whether one specific close-pair pattern persists throughout the infinite prime sequence.[2]

The pair p, p + 2 is part of a wider question: which fixed patterns can consist entirely of primes infinitely often? Some patterns have automatic divisibility obstacles. For p, p + 2, p + 4 with p greater than three, one member is divisible by three, so that triple cannot work. The two-number pattern avoids this particular obstruction. Proving that allowed patterns recur would connect local arithmetic restrictions to the global distribution of primes.[2]