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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"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

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