Results 1 to 5 of 5

Thread: elem#theo - induction + Bertrand's

  1. #1
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101

    elem#theo - induction + Bertrand's

    OK: $\displaystyle p_{n}$ denotes the $\displaystyle nth$ prime. For $\displaystyle n>3$, show that $\displaystyle p_{n} < p_{1} + p_{2} + ... + p_{n-1}$
    the hint was to use induction and Bertrand's conjecture

    ok: so the answer is in the book, but its not helpful b/c I don't get one jump

    so they do this:
    So for n=4, $\displaystyle p_{4} = 7 < 2+3+5$, which is true.
    Assume for some k, $\displaystyle p_{k} < p_{1} + p_{2} + ... + p_{k-1}$
    By Bertrand, there exists a prime between $\displaystyle p_{k}$ and $\displaystyle 2p_{k}$

    Here is where I'm lost: they go from there to $\displaystyle p_{k+1} \leq 2p_{k} < (p_{1} + p_{2} + ... + p_{k_{1}})$,
    thus proving that statement is true for $\displaystyle k+1$

    How? *so confused*
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101
    anyone? please - kinda urgent ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101
    HELP!!!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    by our inductive assumption, $\displaystyle p_{k} < p_{1} + ... + p_{k-1} $

    so add $\displaystyle p_{k} $ to both sides:

    $\displaystyle p_{k} + p_{k} < p_{1} + ... + p_{k} $

    which is the same thing as $\displaystyle 2p_{k} < p_{1} + ... + p_{k} $

    and we know $\displaystyle p_{k+1} \leq 2p_{k} $ by the conjecture, so...

    $\displaystyle p_{k+1} < p_{1} + ... + p_{k} $
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101
    Ah - thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. elem#theo - phi and odd #
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Dec 2nd 2008, 06:45 PM
  2. elem.num.theo - diophantine equations
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: Oct 9th 2008, 02:23 PM
  3. elem.num.theo - GCD
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: Oct 9th 2008, 01:23 PM
  4. elem.num.theo - help w/proof
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Sep 26th 2008, 06:18 AM
  5. elem. # theo - gcd(a, a+n) divides n ...
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Sep 23rd 2008, 08:25 PM

Search Tags


/mathhelpforum @mathhelpforum