GPT-6 Astra has made a major breakthrough in the Goldbach Conjecture

An AI system called Astra produced an 8-page proof of the Liouville version of the Goldbach conjecture, which states that every positive even number greater than 2 is the sum of two positive numbers with Liouville value -1, according to a post by Captain Sude on X. The result was Lean verified, and Astra first proved the unconditional conjecture for multiples of 4 before extending it to the entire conjecture via elementary methods, building on Alexander P. Mangerel's 2024 paper that proved the conjecture for all sufficiently large even integers under the Generalised Riemann Hypothesis for Dirichlet L-functions. A computer search found no counterexamples to the broader claim that every integer of at least 4, even or odd, is the sum of two integers with Liouville value -1 within 10^6.

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?