Results 1 to 5 of 5

Math Help - How do i show this is transendental

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    23

    How do i show this is transendental

    If e-pi is transendental over Q, how can i show that
    e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

    I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Maybe I am wrong but suppose that (e-pi)^3 is not transcental
    It means that (e-pi)^3 is solution of a polynomial equation with integer coefficients

    a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\;=\;0

    Then

    a_n((e-\pi)^3)^n + a_{n-1}((e-\pi)^3)^{n-1} + ... + a_1(e-\pi)^3 + a_0\;=\; 0

    a_n(e-\pi)^{3n} + a_{n-1}(e-\pi)^{(3n-3)} + ... + a_1(e-\pi)^3 + a_0\;=\; 0

    Then e-pi is solution of
    a_nx^{3n} + a_{n-1}x^{(3n-3)} + ... + a_1x^3 + a_0\;=\; 0
    Which is not possible because e-pi is supposed transcendental
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2008
    Posts
    23
    Can anyone check if the above is correct please. Thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by thegarden View Post
    If e-pi is transendental over Q, how can i show that
    e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

    I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
    If e-\pi is transcendental so is (e-\pi)^n,\ n \in \mathbb{N} . Now put n=3 and expand.

    CB
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by thegarden View Post
    If e-pi is transendental over Q, how can i show that
    e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.

    I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
    Thread closed due to OP deleting questions after getting help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using the MVT to show this
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 25th 2010, 11:19 AM
  2. Show that x =
    Posted in the Algebra Forum
    Replies: 8
    Last Post: November 25th 2010, 03:43 AM
  3. Show the sum
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 21st 2010, 03:28 AM
  4. Show that ln5 < 1 + 1/2 + 1/3 + 1/4.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 6th 2009, 01:29 AM
  5. how to show show this proof using MAX
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 14th 2009, 12:05 PM

Search Tags


/mathhelpforum @mathhelpforum