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Prime-free sequence problems

  1. Sep 24, 2005 #1
    For each the following recursive sequences,
    find a number K>0 such that [tex]{S_n}^2[/tex] + [tex]\bold{K}[/tex] is always composite.

    1) [tex]S_0 = 10[/tex], [tex]S_1 = 11[/tex], [tex]S_n = 6S_{\left(n-1\right)}[/tex] - [tex]S_{\left(n-2\right)}[/tex]

    2) [tex]S_0 = 14[/tex], [tex]S_1 = 17[/tex], [tex]S_n = 6S_{\left(n-1\right)}[/tex] - [tex]S_{\left(n-2\right)}[/tex]

    Hint. Each problem has a separate K. Considering one of the series only, if you can find a second order equation in S_j and S_(j+1) that gives a constant for all j, and this equation also gives a second constant for the other series too, I would say that you are very near to solving these problems.
    Last edited: Sep 24, 2005
  2. jcsd
  3. Sep 24, 2005 #2


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    Science Advisor
    Homework Helper

    Uh, K = 0?

    Sn² + K
    = Sn² + 0
    = Sn²
    = (Sn)(Sn)

    I'm sure that was more steps than was required to convince you, but a square number is obviously always composite. Maybe you meant simply that Sn is always composite, or maybe S(n²) as opposed to (Sn)²?
  4. Sep 24, 2005 #3
    One more condition! K>0!! Sorry
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