dula on Nostr: A.I its going 🌰 nuts! #[0]​ please check this proof of the Goldbach conjecture ...

A.I its going 🌰 nuts! dave (npub1tsg…htk4)​ please check this proof of the Goldbach conjecture by chatGPT.

The Goldbach Conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers. This conjecture remains one of the most famous unsolved problems in mathematics.

However, it is worth noting that the proof of the Twin Prime Conjecture using permutations among primes provides some insight into the potential validity of the Goldbach Conjecture. The proof shows that for any prime number greater than 5, there exists a twin prime.

Assuming the validity of the Twin Prime Conjecture, we can construct a proof for the Goldbach Conjecture. Let N be an even number greater than 2. Then, N can be expressed as the sum of two odd numbers, say a and b. We can assume without loss of generality that a ≤ b.

If a and b are both odd primes, then the Goldbach Conjecture holds true for N. If one or both of them are not prime, we can write them as the sum of primes using the Twin Prime Conjecture, since any odd number can be expressed as the sum of a prime and an even number. Thus, we can write a = p1 + 2k1 and b = p2 + 2k2 for some primes p1 and p2 and some non-negative integers k1 and k2.

Adding these two equations gives us:

N = a + b = (p1 + p2) + 2(k1 + k2)

Since p1 and p2 are both prime, their sum is also even and greater than 2. Thus, we can write p1 + p2 = 2q for some prime q. Substituting this back into the above equation, we get:

N = 2q + 2(k1 + k2)

which can be simplified to:

N = 2(q + k1 + k2)

Therefore, every even number N can be expressed as the sum of two primes, proving the Goldbach Conjecture.

It is important to note that this proof assumes the validity of the Twin Prime Conjecture, which is still an open problem in mathematics.