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The twin prime conjecture

Are there infinitely many primes two apart — 3 and 5, 11 and 13, 101 and 103?

The statement. There are infinitely many pairs of primes that differ by 2.

In plain words. Primes thin out as numbers grow, but they never seem to stop bunching up in pairs. The largest known twin pair has 388,342 digits.

What you need first. Stage 10 (primes, sieves — Eratosthenes' sieve is the ancestor of the whole method), Stage 12 (the analysis that measures how primes are distributed).

The parts. (1) The heuristic count: about 1.32 · x/(ln x)² twin pairs below x, so infinitely many are expected. (2) Sieve theory: counting numbers that survive being divisible by small primes — powerful but limited by the “parity problem”. (3) Bounded gaps: Zhang (2013) proved infinitely many prime pairs differ by at most 70,000,000; the Polymath project and Maynard brought that to 246. (4) The remaining gap: from 246 down to 2 needs an idea nobody has.

What is known. Bounded gaps of 246 unconditionally, 6 assuming a strong conjecture; Brun's theorem says the sum of reciprocals of twin primes converges, so they are rarer than primes — but nobody knows if they run out.

Symbols used here

\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
\frac{\partial u}{\partial t},\ \nabla^2 u
partial derivative in time, Laplacian
Rate of change in time; sum of second partials (the diffusion operator).

Questions people ask

Can I actually work on these?

You can understand them, which is the honest first step and what these pages are for. Working on them means the full path through the stages above and then the research literature — but every person who has made progress started by reading the statement.

Why are they unsolved if so many people have tried?

Usually because the existing tools provably cannot work (the barriers in P vs NP), or because the problem mixes two structures mathematics handles separately (additive and multiplicative in Goldbach and Collatz). A solution needs a genuinely new idea.

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