Question

Show that every integer greater than 11 is the sum of two composite integers.

Short answer

If n > 11, it can be written as a sum of two composite numbers.

Step-by-step solution

Step 1

Note that the only triplet of the form k,k+2,k+4which are all primes is when k = 3.

This can be see as follows.

If does not divide kand k + 2, then it should divide k + 1which implies that 3divides k + 4.

Now suppose that a number ncannot be written as a sum of two composite numbers.

Step 2   

This implies that n - 4,n - 6 and n - 8 are all primes (since 4,6,8are composite).

This implies from above that n - 8 = 3which implies that n = 11 .

Thus if n > 11, it can be written as a sum of two composite numbers.