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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 = p

_{1}+ p

_{2})

This implies that...

|n - p

_{1}| = |n - p

_{2}|

«» (n - p

_{1})

^{2}= (n - p

_{2})

^{2}

«» -2np

_{1}+ p

_{1}

^{2}= -2np

_{2}+ p

_{2}

^{2}

«» p

_{1}

^{2}- 2np

_{1}+ np

_{2}- p

_{2}

^{2}= 0

«» p

_{1}

^{2}- 2(n(p

_{1}- p

_{2})) - p

_{2}

^{2}= 0

«» 0 = (p

_{1}+ p

_{2})(p

_{1}- p

_{2}) - 2n(p

_{1}- p

_{2})

«» 0 = (p

_{1}+ p

_{2}- 2n)(p

_{1}- p

_{2})

«» p

_{1}= p

_{2}or

**2n = p**

_{1}+ p_{2}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.