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Math Help - Euclid Numbers

  1. #1
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    Euclid Numbers

    We can define Euclid numbers, e(n), by the following inductive definition:

    e(1) = 2
    e(n) = e(1) *...*e(n-1) + 1, n > 1

    Calculating, we get
    e(1) = 2
    e(2) = e(1) + 1 = 2+1 =3
    e(3) = e(1) * e(2) + 1 = 2*3+1 =7
    e(4) = e(1) * e(2) * e(3) + 1 = 2*3*7+1 =43

    Show that,
    (i) when j /= k ( '/=' is supposed to mean 'NOT equal to!), e(j) and e(k) are
    relatively prime, i.e. gcd(e(j), e(k)) = 1
    (ii) for n > 0,
    e(n+1) = (e(n))2 - e(n) + 1

    No idea how to do (i), tried (ii) using induction but didn't get very far. Please help, would
    be ever so grateful!!!
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  2. #2
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    Re: Euclid Numbers

    (ii) should be done using induction.
    e(n+1) = e(1)*...*e(n-1)*e(n) + 1
    = (e(n)-1)*e(n) + 1
    = e(n)*e(n) - e(n) + 1 which is what you are required to prove.
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  3. #3
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    Re: Euclid Numbers

    Thanks coolge!
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