MathDB

Let

n_{1}, \cdots, n_{k}

and

a

be positive integers which satify the following conditions:[*] for any

i \neq j

(n_{i}, n_{j})=1

, [*] for any

i

a^{n_{i}} \equiv 1 \pmod{n_i}

, [*] for any

i

n_{i}

does not divide

a-1

. Show that there exist at least

2^{k+1}-2

integers

x>1

with

a^{x} \equiv 1 \pmod{x}

Problem Statement