Alternative Form of Goldbach's Conjecture

The Goldbach Conjecture states that every number from 4 onwards can be written as a sum of primes in at least one way. It is equivalent to the statement that every number can be written as a sum of three primes.
For, suppose the Goldbach conjecture holds. Then for  
\[n \ge 4, \; 2n-2 \ge 6\]
  and so can be written as a sum of two odd primes and we can write  
. Then  
\[2n=p_1+p_1+2, \; 2n+1=p_1+p_2+3\]
, showing that  
\[2n, \; 2n+1\]
  can be written as a sum of three primes and all integers from 8 onwards can be written in this form. We can also write
6=2+2+2, 7=2+2+3
Conversely suppose that every integer greater than or equal to 6 can be written as the sum of three primes. Consider the integer  
\[2n, n \ge 3\]
. We know that  
\[p_1, \; p_2, \; p_3\]
  cannot always all be odd, or we could not add them to obtain an even number. Hence one of them must be 2 - suppose  
. If one of  
\[p_2, \; p_3\]
  is 2 then so is the other since  
  is even, contradicting  
\[n \gt 3\]
. Hence  
  is written as the sum of two primes, which is Goldbach's conjecture.

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