A claim announces a complete proof, now Lean-verified, of the Liouville version of Goldbach: every positive even integer greater than 2 can be written as the sum of two positive integers with Liouville value −1 (equivalently, products of an odd number of primes). The timeline presented notes that Alexander P. Mangerel proved the statement for sufficiently large even integers assuming the Generalized Riemann Hypothesis for Dirichlet L-functions in 2024; an AI named Astra reportedly first established the result unconditionally for multiples of 4 and then produced an elementary, eight‑page unconditional proof covering all even integers. The full formalization is said to be available in a Lean theorem‑proving repository.
Specifics emphasized include the concrete algebraic condition (Liouville value −1) and the claim of both a short human‑readable proof and a machine‑checked verification, representing an AI‑driven milestone in number theory. A commenter further reports computational evidence suggesting an even stronger statement - every integer ≥4, even or odd, is the sum of two Liouville‑value −1 numbers - with no counterexamples found up to 10^6. The materials referenced include the 2024 preprint, the eight‑page proof, and a GitHub Lean project purporting to formalize the result.
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