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Math Help - homework problem

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
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    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
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