--- On Wed, 7/30/08, Bernardo Boncompagni <

RedGolpe@...> wrote:

> 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.

There's a well-known equivalent proof for divisors of Fermat Numbers (with an extra stage at the end that you don't need). It's worth understanding that, and then trying to adapt it to arbitrary cyclotomic numbers.

Phil