Loading ...
Sorry, an error occurred while loading the content.

19513Factors of cyclotomic numbers

Expand Messages
  • Bernardo Boncompagni
    Jul 30, 2008
    • 0 Attachment
      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

      ________________________________________________

      "When the missionaries arrived, the Africans had
      the land and the missionaries had the bible.
      They taught how to pray with our eyes closed.
      When we opened them, they had the land and we
      had the bible"
      Jomo Kenyatta

      VisualTaxa - Taxonomy in a visual way
      http://visualtaxa.redgolpe.com
      ________________________________________________
    • Show all 5 messages in this topic