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I think that this is a proof that all even numbers are sum of two odd primes, but would be grateful for anyone who could have a look to see if I missed something

A formal proof will go as follows:- We take two odd primes p1 and p2 So that $p1≥ p2$ $p1+ p2≥ 2p2$ But $p1$ and $p2$ are odd primes and sum of $2$ odd no is always even So $p1+p2=2x$ $2x≥2p2$ $x≥p2 ...(1)$ Now if $p1≥p2$ $p1≥x ...(2)$ combining $(1)$ and $(2)$ $p1≥x≥p2$ Now $p2$ is the smaller prime and $3$ is the smallest prime and there are infinitely many primes so greatest prime is infinite $∞ ≥ x ≥ 3$ So $x$ belongs to $[3,∞]$ , which in turn makes $2x$ every even no starting from $6$. But if sum of two primes is equal to $2x$ then $x$ can also be equal to $2$ $x$ belongs to $2$ to infinity , So $2x$ is every even no greater than $2$. But $2x$ is the sum of two primes $p1$ and $p2$. Every even no greater than $2$ is the sum of two primes. Second case can easily be proved by taking $3,5..$ as the third prime.

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No, that is not a correct proof. –  Tobias Kildetoft Jan 16 '13 at 14:38
Mathematicians have spent 250 years trying to prove it without success. If you spend 25 minutes learning how to format math equations here, someone might try to spend 5 minutes reading your "formal proof" to point out your errors. –  leonbloy Jan 16 '13 at 14:41
@Tobias Write the first flaw in reasoning in an answer so we can upvote it! The user is looking to see if he 'missed something' and that counts. –  rschwieb Jan 16 '13 at 14:42
Not again... math.stackexchange.com/questions/206307/… –  L. F. Jan 16 '13 at 14:42
(1) Write using LaTeX in order to make your mathematics clear. As it stands it's hard to read and understand. (2) Be more thorough and write "numbers" and not a sloppy "no". (3) Use periods, commas and other punctuation marks to make yourself clear. (4) You may want to learn, or at least remember, some basic maths before you engage in such a formidalbe task as trying to prove G.B.: 3 is not "the smallest prime", it is 2. Perhaps you meant "odd prime" and we're back at basic language skills. (5) It's not even wrong to write that "the greatest prime is infinite": it's worse and completely false. –  DonAntonio Jan 16 '13 at 14:46
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1 Answer

In your sentence starting with "Now $p_2$ is the smaller prime...", you do a bit of hand-waving and simply declare that (most of) your desired conclusion holds without any justification whatsoever. In particular, you can conclude that $2x$ is some even number starting at $6$, but cannot conclude (without justification) that every such number is reachable in this way. A proof requires, well, proof.

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Your last sentence reminds me of a quote by a former Prime Minister: A proof is a proof. What kind of a proof? It's a proof. A proof is a proof. And when you have a good proof, it's because it's proven. (J. Chrétien) –  Arthur Fischer Jan 16 '13 at 14:48
@Arthur That is almost at the level of the poetry of Donald Rumsfeld. –  Michael Greinecker Jan 16 '13 at 15:27
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