Question 500690
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1.  Find an even integer greater than 2 that cannot be expressed as the sum of two prime numbers.


2.  Let *[tex \Large \varphi] be an odd integer greater than 5 such that three not necessarily distinct prime numbers that sum to *[tex \Large \varphi] do not exist.  Let *[tex \Large p] be a prime number greater than 2, then *[tex \Large p] must be odd.  The difference of any two odd integers is even *[tex \Large \left((2m\ -\ 1\)\ -\ \left(2n\ -\ 1\right)\ =\ (2m\ -\ 2n)\right)].  Hence, *[tex \Large \varphi\ -\ p] is even.  But Goldberg's conjecture says that *[tex \Large \varphi\ -\ p] must be the sum of two primes, and therefore *[tex \Large \varphi] must be the sum of three primes.  Therefore, if *[tex \Large \varphi] exists, there must be a *[tex \Large \varphi\ -\ p] that is even and not the sum of two primes proving Goldberg's Conjecture false.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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