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Cubic, quartic and higher order residues

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  • spoogie37
    Questions: (1) If 2 and 5 are both cubic residues of a prime p, is 10 a cubic residue please? (2) Does the same apply to quatic and highr order residues? (3)
    Message 1 of 2 , Aug 7, 2006
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      Questions:

      (1) If 2 and 5 are both cubic residues of a prime p, is 10 a cubic
      residue please?

      (2) Does the same apply to quatic and highr order residues?

      (3) If p = 2 mod 3, are ALL numbers cubic residues (this seems to be
      the case for low values of p)?

      (4) What if neither 2 nor 5 are cubic residues - what then is the
      status of 10? It seems to me (based only on low values of p) that
      this means 10 IS a cubic residue, providing it is not already a
      primitive root.

      (5) Do similar statements to (4) apply to quartic and higher order
      residues?

      (6) Where can I find out more please?

      May thanks

      Spoogie
    • Chris Caldwell
      ... If A is a cubic residue, that means there is an a such that a^3 = A. So yes, if A and B are cubic residues (say the cubes of a and b), so is their product
      Message 2 of 2 , Aug 7, 2006
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        > (1) If 2 and 5 are both cubic residues of a prime p, is 10 a
        > cubic residue please?

        If A is a cubic residue, that means there is an a such that a^3 = A.
        So yes, if A and B are cubic residues (say the cubes of a and b),
        so is their product (it is the cube of ab).

        > (2) Does the same apply to quatic and highr order residues?

        Indeed.

        > (3) If p = 2 mod 3, are ALL numbers cubic residues (this
        > seems to be the case for low values of p)?

        The order of the elements divide the order of the group.
        The group of integers prime to p, modulo p, has order p-1.
        So if p is 2 mod three, this order is not divisible by 3
        and hence all values are cubic residues.

        > (4) What if neither 2 nor 5 are cubic residues - what then is
        > the status of 10? It seems to me (based only on low values of
        > p) that this means 10 IS a cubic residue, providing it is not
        > already a primitive root.

        It might, or might not be. 2,5 and 10 are not cubic residues mod 7

        > (5) Do similar statements to (4) apply to quadric and higher
        > order residues?

        Yes, the statement that it might or might not be holds; the higher
        the degree, the less likely it is a residue.

        > (6) Where can I find out more please?

        An elementary number theory textbook.

        Chris.
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