Results 1 to 5 of 5

Math Help - Basic number theory

  1. #1
    Senior Member
    Joined
    Apr 2009
    Posts
    306

    Basic number theory

    I just started self learning basic number theory so please excuse me for any noob questions:

    1. Prove that the fraction is in lowest terms for every positive integer

    2. Let , show that

    3. Prove that consecutive Fibonacci numbers are always relatively prime.

    4. Show that can never be an integer. (I'm thinking of showing converges to a value... or something along those lines, probs wrong heh)

    Can anyone show me how to do these questions? Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    769

    First question

    I would try a little rearrangement with the first question.

    I would factor out n in the numerator into n(n^2 + 2) and add and subtract 1 in the denominator which would transform the denominator into n^4 + 3n^2 + 2 - 1 which factors into (n^2 +1)(n^2 + 2) - 1. Now notice the common factor (n^2 + 2) in both numerator and denominator.

    I'll let you take it the rest of the way.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    769

    With question #4

    As you may know, it's a harmonic series which sums to infinity as n increases without bound. You need to find a representative term for the sum of the first n terms and show that this term can never be an integer no matter what n is.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    For the first one, you want to find polynomials p(n), q(n) having integer coefficients such that

    p(n)(n^3+2n)+q(n)(n^4+3n^2+1)=1.

    For the third one, use the identity f_n^2-f_{n-1}f_{n+1}=(-1)^{n+1}.

    For number 4, do not use wonderboy's advice, as you will not find a closed form for the nth harmonic number. Here is a hint : consider the highest power of 2 less than n, say 2^m. Then there is no other integer between 1 and n which is divisible by 2^m. So what would you get if you put 1+\frac{1}{2}+...+\frac{1}{n} in a fraction having lowest terms, say \frac{p}{q}? (show that p is odd, while q is even!)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    for the second one, hint: [m,n](m,n)=mn
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basic Set Theory
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: August 24th 2011, 06:08 PM
  2. Textbooks on Galois Theory and Algebraic Number Theory
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 8th 2011, 06:09 PM
  3. Basic Number Theory
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: November 2nd 2009, 09:21 PM
  4. Basic Set theory...
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: May 16th 2009, 05:43 AM
  5. Replies: 2
    Last Post: December 18th 2008, 05:28 PM

Search Tags


/mathhelpforum @mathhelpforum