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Re: [PrimeNumbers] Re: multiplicative group mod M: I claim cyclic iff M=odd prime power or M<6.

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  • David Cleaver
    ... I have just downloaded version 2.5.3, which also says: znprimroot(n): returns a primitive root of n when it exists. What I was trying to convey is that
    Message 1 of 6 , Jan 7, 2013
      On 1/7/2013 8:38 AM, djbroadhurst wrote:
      >
      > David Cleaver wrote:
      >
      > > Pari/GP has a function called znprimroot()
      >
      > Which works perfectly, with
      > > GP/PARI CALCULATOR Version 2.5.0 (released)
      > if you will read the friendly manual:
      > > znprimroot(n): returns a primitive root of n when it exists.

      I have just downloaded version 2.5.3, which also says:
      znprimroot(n): returns a primitive root of n when it exists.

      What I was trying to convey is that Pari/GP can also return what looks like a
      valid answer, even though the input does not have a primitive root, ie:

      ? znprimroot(15)
      %1 = Mod(2, 15)

      ? znprimroot(30)
      %2 = Mod(17, 30)

      ? znprimroot(33)
      %3 = Mod(5, 33)

      ? znprimroot(91)
      %4 = Mod(2, 91)

      I have read the fine/friendly manual, ie ?znprimroot. However, this says
      nothing about what happens when the primitive root does not exist. I have found
      several places online that do discuss that the result in those situations will
      be "undefined", like here:
      http://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html

      However, I believe we are in complete agreement that when you want a primitive
      root of numbers that have (at least) one, then you can use the znprimroot()
      function in Pari/GP to find it.

      -David C.
    • djbroadhurst
      ... I imagine that GP returns an element of maximum order. Iff the group is cyclic this is a primitive root, with order eulerphi(n). All of this is perfectly
      Message 2 of 6 , Jan 8, 2013
        --- In primenumbers@yahoogroups.com,
        David Cleaver wrote:

        > What I was trying to convey is that Pari/GP can also
        > return what looks like a valid answer, even though the
        > input does not have a primitive root

        I imagine that GP returns an element of maximum order.
        Iff the group is cyclic this is a primitive root,
        with order eulerphi(n). All of this is perfectly
        consistent with the rubric:

        znprimroot(n): returns a primitive root of n when it exists.

        as we both agree.

        David
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