1. ## homework problem

(a) For any integer n > 2, prove that one can write
n = p+m such that p is a prime number larger than
n/2 and m is an element of Z+. (Hint. You may use Theorem 1.)

(b) Let P = {p|p is prime} U {1}. Prove that for any
n is an element of Z+, there exists r, an element of Z+ such that
n = a1 + a2 + ... + ar
where ai is an element in P for each i, and a1 < a2 < ... < ar.
For example, 4 = 1 + 3; 6 = 1 + 2 + 3; 14 = 3 + 11.
(Hint. For n >= 3, use (a) and strong mathematical
induction).

theorem 1 (as referenced) - For any real number x > 1, there exists a
prime number p satisfying x < p < 2x.

2. For part (a) note that in the theorem you're given you can divide through by two to obtain $\forall x \in \mathbb{R}$ $\exists p$ such that $\frac{x}{2} < p < x$

Can you see how it follows now?

Spoiler:
Choose such a $p$ and let $m = n - p$, clearly m is a postive integer and so you are done.

3. For the second part, observe that you can repeat what you done with $n$ to $m$
So, $\exists q$ such that q is prime and $\frac{m}{2} < q < m$

You should be able to see how the rest follows.

Hope this helped

pomp