MathDB

Goldbach's conjectures

The binary Goldbach conjecture asserts that every even integer greater than 22 is the sum of two primes: ∀n∈Z,  (n>2∧2∣n)⇒∃\forall n\in\mathbb{Z},\; (n>2\land 2\mid n)\Rightarrow\exists primes p,qp,q such that n=p+qn=p+q. The ternary Goldbach conjecture asserts that every odd integer greater than 55 is the sum of three primes: ∀n∈Z,  (n>5∧2∤n)⇒∃\forall n\in\mathbb{Z},\; (n>5\land 2\nmid n)\Rightarrow\exists primes p,q,rp,q,r such that n=p+q+rn=p+q+r.

References

Primary source

Zenodo

Additional references

additive-number-theorynumber-theoryprime-numbers
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Progress summary

Refreshed
Claimed progress

An unverified 2026 manuscript claims both Goldbach conjectures are proved, but only the three-prime version is generally regarded as established and the two-prime version remains unresolved.

The problem comprises the binary and ternary assertions about representing integers as sums of primes. The ternary assertion was proved by Harald Helfgott, while no verified proof of the binary assertion is reported.

Known results

  • Vinogradov, 1937: every sufficiently large odd integer is a sum of three primes.
  • Chen, 1973: every sufficiently large even integer is a prime plus a number with at most two prime factors.
  • Helfgott, 2013–2014: the ternary conjecture was proved, combining explicit estimates with computation.
  • Helfgott and Platt, 2013: extensive numerical verification supported the ternary result.

February 2026 claimed proof and September 2026 repository report

Jason R. South’s manuscript, submitted February 17, 2026, claims proofs of both the binary and ternary conjectures, but no independent verification or acceptance is reported. On September 28, 2026, a repository listing by Daniel Avilés Hurtado announced a three-prime difference theorem connected with Goldbach’s conjectures; its theorem and proof details were unavailable.

Current status (as of September 2026): The ternary conjecture is generally recorded as proved, while the binary conjecture remains open; South’s claimed proof and the new repository result are unverified.

Sources

Solutions 0

No solutions have been posted yet.