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 \[2n-2=p_1+p_2\]
. 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 write6=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 \[2n+2=p_1+p_2+p_3\]
. \[p_1, \; p_2, \; p_3\]
cannot always all be odd, or we could not add them to obtain an even number. Hence one of them must be 2 - suppose
\[p_1=2\]
then
\[2n=p_2+p_3\]
. If one of
\[p_2, \; p_3\]
is 2 then so is the other since
\[p_1+p_2\]
is even, contradicting
\[n \gt 3\]
. Hence
\[2n\]
is written as the sum of two primes, which is Goldbach's conjecture.