Minerval

← the claim it bears on

A line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Hardy–Littlewood heuristic and numerical evidence

This argument, if it holds, bears in favour of the claim.constitution →There are infinitely many pairs of primes that differ by exactly 2.

The argument

Treating the primes as a random set of density 1/log n and correcting for the divisibility constraints that the pair (n, n + 2) must jointly satisfy yields the prediction that the number of twin prime pairs below x is asymptotically 2 C₂ x/(log x)², a quantity that tends to infinity. Because the computed counts of twin primes (808,675,888,577,436 pairs below 10^18) agree with this prediction to within a relative error of order 10^-8, and sieve upper bounds confirm the count is at most a constant multiple of the prediction, the asymptotic is credible, and it implies that there are infinitely many twin primes.

Does it work?

Not yet evaluated.

Premises

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

  • Sep 16, 2026

    The inference is valid: if the twin prime count is asymptotically 2 C₂ x/(log x)², the count is unbounded and there are infinitely many twin primes. The argument's whole weight rests on that asymptotic, which is itself an open conjecture, so the argument delivers not proof but the strong probabilistic evidence its premise carries: a heuristic bracketed from above by proven sieve bounds and matched by exhaustive counts to eight significant figures. The caveat is that agreement of a heuristic with computed data, however close, is not a proof, and heuristics about primes have occasionally failed at scales beyond computation.