The Distance to the Next Prime

The one thing primes always leave behind

Line the primes up in order, 2, 3, 5, 7, 11, 13, and stare at the spaces. One step from 2 to 3, two steps from 3 to 5, then 5 to 7, then a jump of four to 11. Those spaces, the prime gaps, are what is left when you subtract each prime from the next, and they carry more of the primes' strange character than the primes themselves. A prime is just a number. A gap is a piece of the pattern.

The previous number-theory drop wound the number line into a spiral and found lines that had no right to be there. This one does something quieter and, in the end, stranger. It just measures the distances. And out of those distances falls a fact that ought to be a contradiction: the gaps grow without bound, running off to a hundred, a thousand, a million empty numbers in a row, and at the very same time the smallest gap there is, a step of 2, keeps returning forever, or seems to, because nobody on Earth has managed to prove it does.

Everything below is computed live. The page runs a sieve in your browser on load, the same one from the spiral drop, and every gap, every bar, every count is recomputed from scratch. No stored list, nothing to go stale.

The spectrum of gaps

The first console walks the primes up to ten thousand, a hundred thousand, or a million, and tallies how often each gap size appears. The result is not a smear. It is a shape with sharp preferences.

Two things jump out. First, every gap but one is even. The single gap of 1, from 2 to 3, is the only odd distance that ever occurs, because after 2 every prime is odd, and the space between two odd numbers is even. So the primes, past their one even member, march in even-sized strides.

Second, and this is the beautiful part, the histogram combs: every gap that is a multiple of 6 stands taller than its neighbours, and gap 6 towers over everything. It is the single most common distance between consecutive primes, and it stays the champion from the prime 947 all the way up past 10³⁵ before the value 30 finally overtakes it. Why 6? Because a gap of 6 is the easiest one to survive. Step 6 from a prime bigger than 3 and you land on another number that is neither even nor a multiple of 3, the two cheapest ways for a candidate to be killed. Any other small gap forces one endpoint or the other onto a multiple of 2 or 3 more often. It is exactly the small-factor bookkeeping that split Goldbach's comet into bands, here it tunes which gaps the primes prefer.

The twins that won't quit

The smallest gap, 2, has a name and a fan club: the twin primes. 3 and 5, 5 and 7, 11 and 13, 17 and 19, and they never stop appearing, 10,007 and 10,009, 1,000,000,007,000 and its partner, out as far as anyone has ever computed. There are eight twin pairs below 100 alone.

How thick on the ground are they? In 1923 G. H. Hardy and J. E. Littlewood made a conjecture precise enough to check. If primes near x have density about 1/ln x (that is the Prime Number Theorem), then twins, needing two near-misses at once, should have density about 1/(ln x)², nudged by a constant that accounts for the correlation between the two conditions. That constant is the twin-prime constant, C₂ ≈ 0.6601618, and the prediction is that the count of twins up to x, written π₂(x), tracks

π₂(x) ≈ 2·C₂·∫ dt / (ln t)²

The second console sieves to a million, counts the twins for real, and lays them against the guess. The crude version, 2C₂·x/(ln x)², runs low, the same way plain x/ln x undersells the primes. But the integral form shadows the true count to within a percent or two, the exact echo of how li(x) beat x/ln x in the spiral drop. Watch the ratio close on 1 from below as x climbs.

Which means the arithmetic is telling us, loudly, that twin primes should go on forever. The formula has no last term. And yet: the twin prime conjecture, that there are infinitely many, is unproven, one of the oldest open questions in mathematics. What we do have is a thunderclap from 2013. Yitang Zhang, an unknown lecturer, proved that some gap size no larger than 70 million recurs infinitely often, the first time anyone had bounded the gaps at all. Within a year a global collaboration, the Polymath project, ground that bound down to 246. So we know for certain that some gap below 246 comes back forever. We just cannot yet prove that gap is 2.

The gap that grows forever

Now the other direction. The third console hunts record gaps, the first place each new gap size ever appears. Gap 6 debuts after 23, gap 34 after 1,327, gap 36 after 9,551, and by half a million the record stands at a gap of 114, a wall of 113 straight composites yawning after the prime 492,113. The records arrive rarely, and the table scores each by its merit, the gap divided by the average gap near there (ln p), so a merit of 8 means eight ordinary strides of empty number line back to back.

That the gaps grow without any ceiling is, unlike almost everything else here, easy to prove, and the proof is a small marvel. Pick any n. Look at the run

n! + 2, n! + 3, …, n! + n

The first is divisible by 2, the second by 3, and in general n! + k is divisible by k, because k divides n! for every k ≤ n. So all n − 1 of these numbers are composite, a prime-free desert of any width you care to name. Want a million composites in a row? Start at 1,000,001! + 2. Emptiness, it turns out, is cheap.

So here is the shape of the mystery, and it is the whole reason gaps are worth staring at. Set the two facts side by side. The gaps between primes grow without bound, provably, you can manufacture a run of composites as long as you like. And the gap of exactly 2 keeps coming back, apparently forever, though no proof exists. The primes get arbitrarily lonely and never lose the habit of arriving in pairs. Both at once. The number line stretches out into longer and longer stretches of nothing, and threaded through all of it, refusing to thin away, are the twins, 11 and 13, standing two apart the way they did at the very start.

Everything on this page remeasured itself the moment you loaded it. The distances are simple to compute and the pattern in them is plain to see. Whether it goes on forever is a question that has outlasted everyone who ever asked it.