Primes do not repel each other, exactly

The primes thin out forever — the average gap near n grows like the logarithm of n — and yet pairs just two apart keep turning up as far as anyone has looked. Whether they turn up forever is the twin prime conjecture, and the last decade turned it from untouchable into merely unfinished.

The prime number theorem sets the baseline: near n, primes appear with density about 1/ln n, so gaps average around ln n and drift upward without bound. Gaps can also be manufactured as large as you like on demand.¹ The conjecture concerns the opposite extreme — that the minimum gap of 2 recurs infinitely often. The #twin-primes keep obliging: 11 and 13, 101 and 103, on up to a record pair with 388,342 digits. What nobody can yet prove is that the supply never runs dry.

Until April 2013, nobody could prove that any bounded gap recurs infinitely often. Then @Zhang — a fifty-eight-year-old lecturer with almost no publication record — showed that infinitely many prime pairs lie at most 70,000,000 apart.² Seventy million is not two, but the theorem crossed the only line that mattered: from "gaps may simply grow" to "some finite gap recurs forever."

The bound then collapsed in public. Within months Maynard — and Tao, independently — found a more flexible sieve, and the Polymath 8 project crowd-sourced the optimisation down to 246, where it has sat since. Under the unproven Elliott–Halberstam conjecture the same machinery gives 6, and there it jams: the parity problem, sieve theory's known blind spot, stands between 6 and 2. The story of #prime-gaps since 2013 is a wall approached at speed, then touched.

Where the repulsion actually lives

You might expect primes to repel — to space themselves out and avoid crowding, the way samples from many physical processes do. They do not. At local scales the gaps behave like a process with no memory at all: clusters and deserts in just the proportions pure chance predicts, which is Cramér's random model of the primes and the reason the figure above looks the way it does. The twin prime conjecture is, in this reading, simply the assertion that chance keeps winning forever.

The repulsion exists — one level down. In 1973 Montgomery computed the pair correlation of the zeros of the Riemann zeta function and found that the zeros genuinely avoid one another, with a precise deficit of close pairs. Shown the formula over tea at the Institute for Advanced Study, @Dyson recognised it instantly: it is the pair correlation of eigenvalues of large random Hermitian matrices — the GUE statistics that govern energy levels of heavy atomic nuclei.³

The resulting picture is still conjectural, but the numerical evidence — most famously Odlyzko's computations of zeros far up the critical line — is overwhelming, and it resolves the apparent paradox neatly. The primes are free to huddle because the discipline lives elsewhere: the zeros, spaced like the spectrum of some undiscovered operator, enforce the smooth global law of the prime count while local prime behaviour stays honestly random. The connection to #random-matrices is the "repulsion" people intuit — real, quantitative, and one Fourier transform away from the primes themselves. Primes do not repel each other, exactly; the things that orchestrate them do.