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Cyclotomic polynomials' factors - the second coming...

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  • Phil Carmody
    I m curious if there s anything special about the factors of cyclotomic polynomials which are also factors of smaller cyclotomics. e.g. 67 | Phi(1474, 3) but
    Message 1 of 2 , Dec 25, 2002
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      I'm curious if there's anything special about the factors of cyclotomic
      polynomials which are also factors of smaller cyclotomics.

      e.g.
      67 | Phi(1474, 3)
      but
      67 | Phi(22, 3) first.

      These repeated factors do not have the same properties as the primitive
      factors (not ==1 modulo the exponent).

      Is there anything special about them, or are they merely chaff that gets
      discarded as uninteresting, or an annoyace?

      Phil


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    • Gary Chaffey
      ... I m curious if there s anything special about the ... I have also looked at these factors.. It seems and this is only a conjecture that Let a1
      Message 2 of 2 , Jan 2, 2003
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        --- Phil Carmody <thefatphil@...> wrote: >
        I'm curious if there's anything special about the
        > factors of cyclotomic
        > polynomials which are also factors of smaller
        > cyclotomics.
        >
        > e.g.
        > 67 | Phi(1474, 3)
        > but
        > 67 | Phi(22, 3) first.
        >
        > These repeated factors do not have the same
        > properties as the primitive
        > factors (not ==1 modulo the exponent).
        >
        > Is there anything special about them, or are they
        > merely chaff that gets
        > discarded as uninteresting, or an annoyace?
        >
        > Phil
        >
        >
        I have also looked at these factors..
        It seems and this is only a conjecture that
        Let a1<a2
        then if p|a1
        p|a2 => a2=p*a1. (note <= is not true)
        A few examples

        17|phi(272,7) and 17|phi(16,7) also 16*17=272
        113|phi(1582,7) and 113|phi(14,7) also 14*113=1582
        59|phi(1711,7) and 59|phi(29,7) also 29*59=1711
        5|phi(2500,7) and 5|phi(500,7) also 5*500=2500
        109|phi(2943,7) and 109|(27,7) also 27*109=2943

        Gary



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