Proof That Every Integer Greater Than 11 Is The Sum Of Two Composite Integers

An integer \(n\) can be either odd or even. If it is even and greater than 11, then it can be presented as \(4k\) or \(4k+2\), where \(k \ge 3\).

If \(n=4k\), then \(n=2k + 2k\). The only time \(2k\) is prime is when \(k=1\), but we are only considering when \(k \ge 3\), so \(n=4k\) is the sum of composites.

If \(n=4k+2\), then \(n= 2k + 2(k+1)\). Since \(k \ge 3\), then both \(2k\) and \(2(k+1)\) are composite.

What about if \(n\) is odd and greater than 11? Let \(m = n - 9\). Since \(n \gt 11\), then \(m \gt 2\), and since \([\text{odd} - \text{odd} = \text{even}]\), then \(m\) is even. This means \(n-9=2k\), where \(k \gt 2\). Since \(n=2k+9\) and since both 9 and \(2k\) are both composites, then \(n\) is the sum of composites.

Styles

(uses cookies)