1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Number Theory

  1. Sep 12, 2008 #1
    1. The problem statement, all variables and given/known data
    Use proof by contradiction to show that every integer greater than 11 is a sum of two composite numbers.

    3. The attempt at a solution
    The question sounds a bit awkward for me since for the numbers I have tested so far. The number can be consists of 1 prime 1 composite or both primes or both composites. So, it is a headache to prove it by contradiction since every case can be true.

    I have done something but I am not sure whether it can even considered to be a proof o not.
    Assume that m is an integer larger than 11 and consists of 1 prime and 1 composite.
    So, m = P + C
    p = m - C
    If m is an odd integer, 2k+1
    I can always find an odd C (2l+1) such that p is a composite number.
    p = 2k+1-2l-1
    =2(k-l)
    Note that I can choose C such that (k-l)>1
    So, p is not a prime number.

    Similiarly, if m is an even number (2k).
    I can always find an even C (2l) such that p is a composite number.
    p = 2k-2l
    =2(k-l)
    Note that I can choose C such that (k-l)>1
    So, p is not a prime number.

    Hence, contradiction for both cases. The conclusion is for integer>11. I can always find 2 composite number such that their sum is equals to the integer.

    But the problem is I think I can also do the same thing such that when I find an appropriate C, p is a prime number. And also for the case both are primes.
     
  2. jcsd
  3. Sep 12, 2008 #2
    I don't see how you used the fact that m is greater than 11. It does not appear in your calculation.

    I have not done a complete calculation, but I assume you could use the property:
    'all prime numbers above q are of form q#·n + m, where 0 < m < q, and m has no prime factor ≤ q' (q# means the product of the primes up to q)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Number Theory
  1. Number Theory (Replies: 3)

  2. Number Theory (Replies: 2)

  3. Number Theory (Replies: 7)

  4. Number of theory (Replies: 1)

Loading...