GPT-6 Astra provides a new proof and settles the “Riemann-version” Goldbach problem by Liu Wei’er.
The classic Goldbach conjecture requires “two prime numbers.” This variant relaxes it to integers for which the total number of prime factors is odd. Alexander P. Mangerel, a mathematician at Durham University, had previously been able to prove the statement only under two conditions: the generalized Riemann hypothesis holds, and the even number is sufficiently large. Astra removes both restrictions and proves that it holds for all even numbers greater than 2: first assume some even number cannot be decomposed, then derive a contradiction.
The complete proof has been written in Lean 4 and compiles correctly; an independent audit repository also reproduces it successfully, with no “sorry” and no additional mathematical axioms. The classic Goldbach conjecture itself is still unresolved—the two addends here may still be composite numbers.
The classic Goldbach conjecture requires “two prime numbers.” This variant relaxes it to integers for which the total number of prime factors is odd. Alexander P. Mangerel, a mathematician at Durham University, had previously been able to prove the statement only under two conditions: the generalized Riemann hypothesis holds, and the even number is sufficiently large. Astra removes both restrictions and proves that it holds for all even numbers greater than 2: first assume some even number cannot be decomposed, then derive a contradiction.
The complete proof has been written in Lean 4 and compiles correctly; an independent audit repository also reproduces it successfully, with no “sorry” and no additional mathematical axioms. The classic Goldbach conjecture itself is still unresolved—the two addends here may still be composite numbers.