The primes refuse to reveal a formula. So in 1936 the Swedish mathematician Harald Cramér tried a wonderfully cheeky move: if the primes behave like a random set, why not just model them as one and see what the randomness predicts? Toss a biased coin for every integer, keep the ones that come up "prime", and study the resulting fake primes. The astonishing payoff is that this crude cartoon reproduces almost everything we know about real primes — the counting function, the typical gap, even a razor-sharp guess for the largest gap — and it does so from a single line of probability.

It is also a cautionary tale. The model is a heuristic, not a theorem, and precisely where it is most confident — the fine structure of primes in short intervals — reality quietly breaks its promise. This page builds the model, cashes out its predictions, and then shows the two famous places it goes wrong.

The model in one line

Recall from the distribution of primes that a number near x is prime with density about 1/\ln x. Cramér turns that density into an actual probability and declares each integer's primality an independent coin toss.

That is the whole engine. It knows nothing about divisibility, factorisation, or the number 2 — it only knows a coin that grows steadily fairer to land on "composite" as n grows. Everything below is just a consequence of that one assumption.

Prediction 1: the count is \operatorname{Li}(x)

How many model-primes are there up to x? By linearity of expectation, the expected count is just the sum of the individual probabilities:

The model doesn't merely reproduce the crude estimate x/\ln x — it lands directly on the logarithmic integral \operatorname{Li}(x), which is the best smooth approximation to \pi(x) and the true main term of the Prime Number Theorem. A one-line probability calculation recovers a theorem that took a century and complex analysis to prove. That is why number theorists take the cartoon seriously.

Prediction 2: the typical gap is \ln p_n

Fix a model-prime near x. The next integer is prime with probability \approx 1/\ln x, the one after that independently the same, and so on — a geometric wait. The expected number of tosses until the next success is the reciprocal of the success probability:

So the model predicts a typical gap of \ln p_n after the n-th prime — exactly the average spacing the real primes obey. It goes further: a geometric wait means the gaps should be exponentially distributed, so a gap of size \lambda \ln p should occur with relative frequency about e^{-\lambda}. Plotting real prime gaps against e^{-\lambda} gives a strikingly good fit — the primes really do look, at this level of detail, like a random exponential process.

Prediction 3: the maximal gap is (\ln x)^2

Here is where the model earns its fame. Typical gaps are \ln p, but how big is the largest gap you should expect to see among all primes up to x? In a run of about x/\ln x independent geometric waits, the longest run of consecutive "composite" tosses is, by an extreme-value estimate, about (\ln x) times the typical gap. Carefully done, the expected maximal gap comes out as (\ln x)^2. This is Cramér's conjecture.

This is one of the boldest conjectures in number theory: a specific constant, 1, pinned to a heuristic. It remains completely open — the best proven upper bound on gaps (from the theory of the zeta zeros) is far weaker, of the order p_n^{0.525}. Yet the data tracks (\ln p)^2 beautifully.

Seeing it: record gaps chase (\ln x)^2

A record (or maximal) gap is one bigger than every gap before it. Sieving the primes up to five million and tracking the running-largest gap gives the staircase below; the smooth curve is Cramér's predicted scale (\ln x)^2. The record gaps climb in lockstep with the curve — always a little beneath it, at roughly 0.7–0.9 times its height, exactly as the record data does out to the largest gaps ever computed.

The useful summary statistic is the merit of a gap, M = g_n / \ln p_n — the gap measured in units of the local average spacing. Cramér's conjecture says merits can grow all the way up to \ln p_n. The table shows merit creeping upward exactly as that predicts:

Worked example: prime gaps near 10^{18}

Let's put the model to work at a genuinely large scale, x = 10^{18}, and then check it against real record data. First the local average spacing:

Typical gap. The model predicts consecutive primes near 10^{18} are spaced, on average, about 41 apart — so out of every 41 or so numbers of that size, one is prime.

Maximal gap. Cramér's conjecture predicts the largest gap seen up to 10^{18} should be about

Reality check. The largest known prime gap near this range is g = 1476, following the prime 1{,}425{,}172{,}824{,}437{,}699{,}411 \approx 1.43\times10^{18}. Its merit is 1476 / 41.8 \approx 35.3. Compare the prediction: 1476 is about 0.86 \times 1718 — the same order of magnitude, sitting just under the predicted ceiling. The model nailed the scale of the biggest gap across eighteen orders of magnitude, from a coin toss. The typical gap \approx 41 is likewise spot-on.

The blind spot: the model ignores divisibility

The model's fatal simplification is that it treats every integer alike — but real primes obey divisibility rules the coins never heard of. The cleanest place to see the damage is twin primes. In the model, "n prime" and "n+2 prime" are independent, so

which predicts \pi_2(x) \sim x/(\ln x)^2 twin pairs up to x. But the true count (the Hardy–Littlewood conjecture, and what the data shows) is

with the correction factor 2C_2 \approx 1.3203. The naive model is off by 32%. The missing factor is the singular series \mathfrak{S}, which accounts for divisibility by small primes: real primes are all odd (so n and n+2 being odd is correlated, not independent), never divisible by 3, and so on. Each small prime nudges the true probability away from the naive product. You meet C_2 properly in Brun's sieve and the twin prime constant.

The refined Cramér–Granville model

Andrew Granville (1995) patched the blind spot. Instead of tossing a coin for every integer, first sieve out the multiples of small primes p \le z, and toss coins only for the survivors — but with a correspondingly larger probability, so the expected count still matches \operatorname{Li}(x). The survivors carry the divisibility structure the naive model threw away, and the singular series \mathfrak{S} reappears automatically. Twin primes then come out with the right constant 2C_2.

The refinement isn't just cosmetic. Granville showed his model predicts

which is bigger than Cramér's constant 1. Divisibility lets primes cluster and thin out more extremely than pure coins allow, so the true record gaps may exceed (\ln x)^2 after all. Whether the constant is 1, 2e^{-\gamma}, or something else is one of the sharpest open questions about prime gaps.

Maier's theorem: the model provably lies in short intervals

The most beautiful refutation of the naive model is Maier's theorem (Helmut Maier, 1985). Count primes in a short interval of length (\ln x)^A — a window whose width is a fixed power of the logarithm. The Cramér model says such an interval should contain about (\ln x)^A / \ln x = (\ln x)^{A-1} primes, with the small, Poisson-sized fluctuations of independent coins. Maier proved this is false.

The mechanism, again, is divisibility. Maier used the small primes (via the sieve) to find windows where an unusual pile-up or drought of primes is forced — a correlation the independent coins can never reproduce. So the Cramér model is not merely imprecise in short intervals; it is provably wrong there. It remains an indispensable guide to the scale of prime-gap phenomena, but its finest predictions must be taken with a fistful of salt.