Goldbach Conjecture

[Enola Straight]

IIRC, Goldbach's Conjecture states that any even number larger than 2 is the sum of two primes.

This is unproven.

Really?

All prime numbers larger than 2 are odd
Odd number + even number = odd number
Even number + even number = even number
Odd number + odd number = even number

Therefore,

(P>2)+(P>2)=even number!

RIGHT?

On another board, a member replies that only proves the sum of two primes have to be even,
NOT that every even number larger than 2 is the sum of two primes.

What's the difference?

[Surlethe]

You're entirely mixed up. Just because all primes larger than two are odd does not mean that all odd numbers larger than two are prime; this is an elementary logic error known as affirming the consequent.

[Enola Straight]

I'm not saying all odds larger than 2 are prime...I'm saying all prime's larger than 2 are odd.

[Surlethe]

No, but your proof is implicitly assuming that all odds larger than 2 are prime. Or it's assuming that since every prime greater than two sums to an even number, every even number is the sum of two odd primes, which is an example of precisely the same logical fallacy.

[Jaepheth]

Take set A to be personal pronouns and set B to be infinitive verbs.

Any pairing of A followed by B is a sentence. (I write., We run., etc.)

But not all sentences are of the form (A, B)

[Kuroneko]

On another board, a member replies that only proves the sum of two primes [>2] have to be even, NOT that every even number larger than 2 is the sum of two primes.

That's exactly right.

What's the difference?

It's the difference between "I see all I eat" and "I eat all I see". They're converses.

[Samurai Rafiki]

Your thought process does prove that the sum of two primes over 2 must be even, but the burden then is to prove that all even numbers over 2 can be formed as the sum of two primes and to prove this indefinitely enough that it can be carried on to infinity. For instance, it's very easy to show that 4, 6, 8, 10, and 12 can be reached as the sum of two primes, but what about 496,187,236,498,763,875,698,176,241,872? Prime numbers decrease in frequency as you go away from zero, so you'd have to find two relatively large prime numbers to sum up to this one. Could you do it eventually? Probably, but I can invent long, high numbers all day, and it's up to you to find primes that high (difficult) and to find the right combination of primes that high to equal my long, even number (extremely difficult). And thus far, no mathematician has been able to do it with indefinite variables.

[Serafina]

Basically, the problem is (IIRC): We do not have a formula to produce primes.

If we had such a formula, we could proove the goldbach conjecture. Without such a formula, it is (nearly?) impossible - because we do not know all primes, and have no way to produce them.
Kinda similar to π - we can calculate more and more of it, but we do not have an "accurate formula" to derive π from anything else.

[Zadius]

Basically, the problem is (IIRC): We do not have a formula to produce primes.

If we had such a formula, we could proove the goldbach conjecture. Without such a formula, it is (nearly?) impossible - because we do not know all primes, and have no way to produce them.

I don't see why such a formula would necessarily be required, though it would probably make it easier to prove/disprove Goldbach.