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Math Help - congruence

  1. #1
    Senior Member Shanks's Avatar
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    congruence

    Show that for any positive integer n, there exist a prime number p and another positive integer m such that:
    (1) p is a prime numer of form 6k+5;
    (2) n is not a multiple of p;
    (3) n-m^3 is a multiple of p.
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  2. #2
    MHF Contributor undefined's Avatar
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    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

    1^3\equiv1\pmod{11}

    2^3\equiv8\pmod{11}

    \dots

    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; September 10th 2010 at 06:12 AM.
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  3. #3
    Senior Member roninpro's Avatar
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    We can rephrase this in the language of congruences:

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

    With this in mind, it suffices to show that the map f:\mathbb{Z}_p\to \mathbb{Z}_p defined by f(x)=x^3 is a bijection. Can you do this?
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