19513Factors of cyclotomic numbers
- Jul 30, 2008I 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.
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