Proven approximations to the conjecture
The argument
Proven results come close to the conjecture from several directions: every odd number greater than five is a sum of three primes, every sufficiently large even number is a prime plus a number with at most two prime factors, and the even numbers that are not sums of two primes form a set of density zero. Since each weakening has been established without meeting any obstruction pointing toward a counterexample, they raise confidence in the binary conjecture, though none of them implies it.
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 →Every odd number greater than five is the sum of three primes.
- 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 →Every sufficiently large even number is the sum of a prime and a number with at most two prime factors.
- 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 →Almost all even numbers are the sum of two primes, in the sense that the exceptional set has density zero.
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
All three premises are established theorems: the ternary Goldbach theorem, Chen's theorem, and the density-zero exceptional set theorem. The inference is supportive only. None of them implies the binary conjecture, and the methods behind them face the parity barrier at exactly the step the full conjecture would need. They show the conjecture is consistent with everything proven nearby, which is corroboration, not proof.
- Sep 25, 2026
All three premises are established theorems: the ternary Goldbach theorem (Vinogradov, completed by Helfgott), Chen's theorem, and the density-zero exceptional set theorem. The inference is supportive only: none of them implies the binary conjecture, and the methods behind them (the circle method and sieves) are known to face the parity barrier at exactly the step the full conjecture would need. They show the conjecture is consistent with everything proven nearby, which is corroboration, not proof.