(Not a Proof of) Goldbach's Conjecture

middleCmusic

I was doing some thinking about good old Goldbach and noticed that every pair of primes was the same distance away from half of the even number.

For example,

3 + 5 = 8 and |4 - 3| = 1 = |4 - 5|
3 + 13 = 16 and |8 - 3| = 5 = |8 - 13|
7 + 17 = 24 and |12 - 7| = 5 = |12 - 17|
19 + 23 = 42 and |21 - 19| = 2 = |21 - 23|
23 + 41 = 64 and |32 - 23| = 9 = |32 - 41|

Now, this is not anything surprising. It's pretty obvious if you just think of the number, say 18 and divide it by two (9 + 9 = 18) and realize that to get other pairs that sum to 18, you simply subtract from the first and add to the second...

9, 9 --> 8, 10 --> 7, 11 (primes) --> 6, 12 --> 5, 13 (primes) --> 4, 14 --> 3, 15 --> 2, 16

Anyway, that's enough introduction. This is not a proof, but I wasn't sure if there was something novel about it or if I was simply fooling around with algebra:

Goldbach's conjecture supposes that for any even number 2n, there are two primes that sum to it ( 2n = p₁ + p₂ )

This implies that...

|n - p₁| = |n - p₂|
«» (n - p₁)² = (n - p₂)²
«» -2np₁ + p₁² = -2np₂ + p₂²
«» p₁² - 2np₁ + np₂ - p₂² = 0
«» p₁² - 2(n(p₁ - p₂)) - p₂² = 0
«» 0 = (p₁ + p₂)(p₁ - p₂) - 2n(p₁ - p₂)
«» 0 = (p₁ + p₂ - 2n)(p₁ - p₂)
«» p₁ = p₂ or 2n = p₁ + p₂

Does the fact that you get back the Conjecture mean anything or is it meaningless because a) it doesn't really prove the prime part of the argument or b) it was obtained by assuming the conjecture in the first place sort of... Perhaps there would be some way to use induction to our advantage here to actually prove it?

Sorry if this was a waste of your time - I'm new at this game.

lpetrich

It does not tell us anything about the validity of Goldbach's conjecture, because it does:
Any 2 even primes or 2 odd primes -> sum is an even number

While his conjecture is
Any even number >= 4 = sum of 2 primes

A implies B
is not equivalent to
B implies A

If a positive integer is a prime other than 2, then it is odd
is not equivalent to
If a positive integer is odd, then it is a prime other than 2

If/then -- implication
false -> false -- true
false -> true -- true
true -> false -- false
true -> true -- true

False can imply true, but true cannot imply false.

robert80

Interesting... Does anybody know if the Goldbach conjecture was proven for any other infinite subset of even numbers, or it is still just the obvious 2*p? was it proved for 2*p with different combination, other than p + p? Thank you in advance.

TylerH

Wow, this is an interesting discovery. Certainly not a waste of my time. I will be doing some thinking, but I have nothing to contribute yet. Thanks for sharing.

TylerH

You may be able to use this to prove every number can be written as the arithmetic mean of primes, which would prove Goldbach's by inclusion.

dhessler

Last time I checked there is only one even prime number... If a number is even and greater than 2 it cannot be prime. Though Petrich is right. Proving Sum -> Even is easy. Even -> Sum is not.

lpetrich

That's right about even primes, though my statement is still correct. I will concede that its wording may have been misleading.