Goldbach's Conjecture
A famous unsolved conjecture in number theory, proposed by mathematician Christian Goldbach in 1742, asserting that every even integer greater than 2 can be expressed as the sum of two prime numbers. The conjecture has not been fully proven to date.
Timeline (last 90 days)
AI has not truly cracked the Goldbach Conjecture; it has only proved a weakened necessary-condition version based on the Liouville function.
GPT-6 Astra unconditionally proved a weakened version of the Goldbach Conjecture using a mathematics paper of only two pages.
The weakened proposition asks: can any even number greater than 2 necessarily be written as the sum of two numbers whose Liouville value is -1?
The classical Goldbach conjecture requires decomposing an even number into the sum of two primes, whereas the Liouville weakened version only requires that both addends have an odd total number of prime factors (Liouville function value of -1), allowing composite numbers to qualify.
The proof written by GPT-6 Astra addresses not the classical Goldbach conjecture, but a weakened Liouville version of it.
Sieve methods are inherently unable to distinguish between numbers with an odd versus even count of prime factors, which is why the Goldbach conjecture is difficult to resolve using pure sieve methods.
In a letter written to Euler in 1742, Goldbach proposed the conjecture that every even number greater than 2 can be expressed as the sum of two prime numbers.
What was solved this time is a variant of the Liouville–Goldbach conjecture, not the original Goldbach's conjecture itself.
Chinese mathematician Chen Jingrun proved the "1+2" form of Goldbach's conjecture (a prime plus the product of two primes).
All Facts (9)
The weakened proposition asks: can any even number greater than 2 necessarily be written as the sum of two numbers whose Liouville value is -1?
50%UnverifiedAI has not truly cracked the Goldbach Conjecture; it has only proved a weakened necessary-condition version based on the Liouville function.
50%UnverifiedThe classical Goldbach conjecture requires decomposing an even number into the sum of two primes, whereas the Liouville weakened version only requires that both addends have an odd total number of prime factors (Liouville function value of -1), allowing composite numbers to qualify.
50%UnverifiedThe proof written by GPT-6 Astra addresses not the classical Goldbach conjecture, but a weakened Liouville version of it.
50%UnverifiedSieve methods are inherently unable to distinguish between numbers with an odd versus even count of prime factors, which is why the Goldbach conjecture is difficult to resolve using pure sieve methods.
50%UnverifiedChinese mathematician Chen Jingrun proved the "1+2" form of Goldbach's conjecture (a prime plus the product of two primes).
50%UnverifiedWhat was solved this time is a variant of the Liouville–Goldbach conjecture, not the original Goldbach's conjecture itself.
50%UnverifiedIn a letter written to Euler in 1742, Goldbach proposed the conjecture that every even number greater than 2 can be expressed as the sum of two prime numbers.
50%UnverifiedGPT-6 Astra unconditionally proved a weakened version of the Goldbach Conjecture using a mathematics paper of only two pages.
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