Results 1 to 3 of 3

Thread: congruence

  1. #1
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374

    congruence

    Show that for any positive integer $\displaystyle n$, there exist a prime number $\displaystyle p$ and another positive integer $\displaystyle m$ such that:
    (1) $\displaystyle p$ is a prime numer of form $\displaystyle 6k+5$;
    (2) $\displaystyle n$ is not a multiple of $\displaystyle p$;
    (3) $\displaystyle n-m^3$ is a multiple of $\displaystyle p$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Edit: nevermind, didn't think it through.

    Edit 2: Here are some thoughts.

    p being congruent to 5 mod 6 is the same as saying p is an odd prime congruent to 2 mod 3.

    Here is an experiment: Note that

    $\displaystyle 1^3\equiv1\pmod{11}$

    $\displaystyle 2^3\equiv8\pmod{11}$

    $\displaystyle \dots$

    $\displaystyle 10^3\equiv10\pmod{11}$

    produces all equivalence classes mod 11 besides zero.

    If this is true for all odd primes congruent to 2 mod 3 and we can prove it, then the only other piece we'd need is that there exist an infinitude of such primes.
    Last edited by undefined; Sep 10th 2010 at 06:12 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    We can rephrase this in the language of congruences:

    (1) $\displaystyle p\equiv 5\pmod{6}$
    (2) $\displaystyle n\not \equiv 0\pmod{p}$
    (3) $\displaystyle n\equiv m^3\pmod{p}$ has a solution

    With this in mind, it suffices to show that the map $\displaystyle f:\mathbb{Z}_p\to \mathbb{Z}_p$ defined by $\displaystyle f(x)=x^3$ is a bijection. Can you do this?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Congruence
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: May 12th 2009, 10:19 AM
  2. Congruence
    Posted in the Geometry Forum
    Replies: 1
    Last Post: Nov 10th 2008, 01:52 PM
  3. congruence
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 30th 2008, 05:04 PM
  4. Congruence
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 30th 2008, 09:11 AM
  5. Congruence
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 29th 2008, 10:57 AM

Search Tags


/mathhelpforum @mathhelpforum