How would you prove every even number is the sum of two prime numbers?

Well, I dont consider 2 to be a counter example. Not adequately so. 2 = 1+1, yes, but whether or not you count 1 among the primes is for the most part arbitrary, a historical artifact, frequently contextual. So many times in some contexts Ive seen people say "all primes and 1" to describe a set, which is as easy to say as "all primes except 1" in any other context. Look back through history and 1 has been considered prime for a long, long time, and the math is equally valid. I have no quarrel considering 1 as prime in this particular problem.

Find me a trivial counter example other than 2=1+1 and it would be sufficient disproof of the claim for me. But if 2=1+1 is the one and only counter example then I say the claim stands true.

After all, all you would have to do to satisfy the "1 is not prime" whiners is to say "every even number greater than 2 is the sum of two primes", artificially limiting the set of consideration, which by the way is as arbitrary as artificially increasing or decreasing the set of primes with the inclusion or exclusion of 1. Whats the functional difference?

But people insist... with a fanaticism. Appeal to mathematical dogmatism. Dont get me wrong, I understand the importance of rigor and precision, and why we need definition. In most cases I wouldnt argue it. But if you truly understood what a prime is, how it behaves, the history of its discovery, the debate around it, you might start to wonder, too, why it shouldnt be considered a prime... at least in some context.

In prime factorization, the ad nauseum factoring of 1 is just a stupid maneuver, like the ad nauseum subtraction of 0. An individual simply would not do it. One would not declassify 1 as prime for the same reason one would not declassify 0 as an integer.

The behavior we are interested in, in this context, is the additive qualities of primes, not its multiplicative properties.

I hate arguing the primality of 1 with people. Its a matter of philosophy and context more than it is about anything innate about mathematics. I submit to you that anything "defined" is manmade and artificial, and shouldnt be treated overly rigidly; such things can be redefined, and refined, and even broadened. As it indeed has been with primeness.

I dont believe the claim has be proven or disproven. Not yet. Its called Goldbachs conjecture. Goldbach claimed that all integers (even or odd) greater than 5 were the sum of three primes. Euler realized that the claim was equivalent to all even integers greater than 2 being the sum of two primes. Again, I dont think its been proven or disproven.

If you allow 1 to be prime in context, the Goldbach conjecture can be broadened to all integers n>1, which strengthens the conjecture, with no loss.

2 = 1+1. But 1 is not a prime number.

It's trivially false.

Hint: you missed an important part of Goldbach's conjecture!

Did you try 2?

its easier to have somebody try disprove it