4.II.8B

Suppose that $a, b$ are coprime positive integers. Write down an integer $d>0$ such that $a^{d} \equiv 1$ modulo $b$. The least such $d$ is the order of $a$ modulo $b$. Show that if the order of $a$ modulo $b$ is $y$, and $a^{x} \equiv 1$ modulo $b$, then $y$ divides $x$.

Let $n \geqslant 2$ and $F_{n}=2^{2^{n}}+1$. Suppose that $p$ is a prime factor of $F_{n}$. Find the order of 2 modulo $p$, and show that $p \equiv 1$ modulo $2^{n+1}$.

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