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19513Factors of cyclotomic numbers

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  • Bernardo Boncompagni
    Jul 30, 2008
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      I have noticed that all factors of a cyclotomic number Phi(n,b) (that is
      the n-th cyclotomic polynomial computed in the point b) are either a
      divisor of n or congruent to 1 modulo n. For example:

      Phi(13,15) = 139013933454241 = 53 * 157483 * 16655159
      All 3 factors are congruent to 1 modulo 13.

      Phi(20,12) = 427016305 = 5 * 85403261
      5 is a divisor of 20 and 85403261 is congruent to 1 modulo 20.

      So I have a few questions:
      - is this a general pattern, or is it just another instance of the "law
      of small numbers"?
      - if so, what is the smallest known counterexample?
      - if not, can anyone point me to a (possibly online) demonstration?

      Thank you for your interest.

      Bernardo Boncompagni


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