# GPT-6 Astra has made a major breakthrough in the Goldbach Conjecture > Source: > Published: 2026-09-23 10:36:59+00:00 Captain Sude on X: "The Liouville version of the Goldbach conjecture is now fully proven and Lean verified! Every positive even number greater than 2 is the sum of two positive numbers with Liouville value -1. https://t.co/hzeoRu6QBy" The Liouville version of the Goldbach conjecture is now fully proven and Lean verified! Every positive even number greater than 2 is the sum of two positive numbers with Liouville value -1. The Liouville version of the Goldbach conjecture is now fully proven and Lean verified! Every positive even number greater than 2 is the sum of two positive numbers with Liouville value -1. Goldbach's conjecture states that every even natural number greater than 2 is the sum of two prime numbers. Goldbach's conjecture for the Liouville function is a natural weakening of the famous conjecture, it just asks the summands to be products of an odd number of primes. In 2018, a mathoverflow user asked this question: mathoverflow.net/questions/3074… In a paper from 2024 Alexander P. Mangerel proved this conjecture for all sufficiently large even integers under the Generalised Riemann Hypothesis for Dirichlet L-functions. arxiv.org/abs/2404.12117 esterday, Astra proved the unconditional conjecture for multiples of 4, building on the previous work by Mangerel. Today, Astra proved the entire conjecture via elementary methods. The proof is 8 pages long. It seems every integer >=4, even or odd, is the sum of two integers with Liouville value -1. A simple computer search found no counterexamples within 10^6. Maybe you can solve this one?