MathDB

Let

n=p_1p_2\cdots p_s

, where

p_1,\ldots,p_s

are distinct odd prime numbers. (a) Prove that the expression

F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},

where the product goes over all subsets

\{p_{i_1},\ldots,p_{i_k}\}

\{p_1,\ldots,p_s\}

(including itself and the empty set), can be written as a polynomial in

x

with integer coefficients. (b) Prove that if

p

is a prime divisor of

F_n(2)

, then either

p\mid n

n\mid p-1

Problem Statement