(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.