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Homework Help: Prime division # Theory

  1. Feb 28, 2015 #1
    1. The problem statement, all variables and given/known data
    If p and q are both greater than or equal to 5, prove that 24|p^2 - q^2

    2. Relevant equations

    3. The attempt at a solution
    24 = 2^3 * 3.
    If p = q = 5, then 24|0.
    If p = 7, q = 5, then 24|24.
    Any other combination, p^2 - q^2 will be greater than 24. I want to show that p^2 - q^2 will always have a prime factor of either two or three, hence it will be divisible by 24.

    If p and q are both odd, p^2 - q^2 will always be even, hence have a two in it's prime factorization.
    A similar situation occurs when p and q are both even.

    However, when p is odd and q is even, p^2 - q^2 is odd. I want to show that in this case, p^2 - q^2 has a three in it's prime factorization. I checked a few examples on wolfram and they all worked out, can't think of a way to prove this though. Anyone have any gentle guidance :D?
  2. jcsd
  3. Feb 28, 2015 #2


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    Science Advisor

    p2 - q2 = (p + q)*(p - q)
  4. Feb 28, 2015 #3
    Thanks... not to mention p and q can never be even since they are primes greater than 5... >.< lol thanks though.
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