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Math Help - An induction problem with primes

  1. #1
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    An induction problem with primes

    Let p_{n} denote the nth prime. For n>3, show that p_{n} < \sum ^{n-1}_{1} p_{i}

    My proof so far:

    Let S = \{ n \in N : p_{n} < \sum ^{n-1}_{1} p_{i} , n>3 \}

    Now, 4 is in S because  p_{4}=7 < 2+3+5 =10

    Assume k in S, then  p_{k} < \sum ^{k-1}_{1} p_{i}

    Now,  \sum ^{k}_{1} p_{i} = \sum ^{k-1}_{1} p_{i} + p_{k} < p_{k}+p_{k} = 2p_{k}

    By the Bertrand's postulate, there exist at least one prime number q such that  p_{k} < q < 2p_{k}

    Pick q to be the smallest prime from the set, then q = p_{k+1} < 2p_{k}

    But now I'm stuck, I'm trying to show that  \sum ^{k}_{1} p_{i} < q , how do I go about doing that?
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  2. #2
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    Do you agree that p_{n+1} < 2p_n = p_n + p_n ---> this is Betrand's conjecture (and it is not easy to prove, but it seems you are allowed to use it)?

    Then by induction,
    p_n + p_n < p_1+...+p_{n-1}+p_n.
    Thus, p_{n+1} < p_1+...+p_{n}.
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