Numerical verification and the Hardy–Littlewood heuristic
The argument
Because the conjecture has been checked by computer for every even number up to 4 × 10^18, any counterexample would have to lie beyond that bound; and because the Hardy–Littlewood heuristic predicts that the number of two-prime representations of an even number grows without bound, with observed counts matching the prediction, a counterexample at larger sizes would require the representation count to collapse to zero against a rapidly growing expectation, which makes the conjecture highly probable.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →Goldbach's conjecture has been verified by computation for all even numbers up to 4 × 10^18.
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →The number of representations of an even number as a sum of two primes tends to infinity as the number tends to infinity.
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 25, 2026
The argument is inductive, not deductive: it makes the conjecture highly probable but cannot prove it. Its computational half rests on the verification up to 4 × 10^18, a refereed and independently double-checked result that is not in doubt. Its heuristic half rests on the representation count tending to infinity, which is itself unproven and follows from the conjectural Hardy–Littlewood asymptotic, so it counts only as plausibility. The caveat is that heuristics of this kind have occasionally misled elsewhere in number theory.
- Sep 25, 2026
The argument is inductive, not deductive: it makes the conjecture highly probable but cannot prove it. Its computational half rests on the verification up to 4 × 10^18, a refereed and independently double-checked result that is not in doubt. Its heuristic half rests on the representation count tending to infinity, which is itself unproven and follows from the conjectural Hardy–Littlewood asymptotic; it carries weight as plausibility, and the caveat is that heuristics of this kind have occasionally misled elsewhere in number theory.