Originally Posted by

**CaptainBlack** ImPerfectHackers proof is quite simple (and neat):

Suppose that there is an integer $\displaystyle N> 11$ not the sum of two composite integers.

Then $\displaystyle N$ is even or odd.

Case 1: $\displaystyle N$ even, put $\displaystyle n_1=4,\ n_2=N-4$, then both $\displaystyle n_1$ and $\displaystyle n_2$ are even and greater than $\displaystyle 2$ and hence composite, but this contradicts our assumption so $\displaystyle N$ cannot be composite.

Case 2: $\displaystyle N$ odd, put $\displaystyle n_1=9,\ n_2=N-9$, then both $\displaystyle n_1$ is composite and $\displaystyle n_2$ is even and greater than $\displaystyle 2$ and hence composite, but this contradicts our assumption so $\displaystyle N$ cannot be composite.

Case 1 and Case 2 together contradict the original assumption and so the theorem: Every integer > 11 is the sum of two composite integers; is proven by contradiction.

RonL