1. ## Index Arithmetic Questions

1) Show that if p is an odd prime and r is a primitive root of p, then $\displaystyle ind_{r}$(p-1) = (p-1)/2

2) Prove that there are infinitely many primes of the form 8k+1.

(Hint: Assume that p1, p2, ..., pn are the only primes of this form. Let Q = (2p1, p2....pn)^k + 1. Show that Q must have an odd prime factor different than p1, p2, ...., pn, and must be of the form 8k + 1.)

2. For the first Q.

$\displaystyle r$ is a primitive root of $\displaystyle p$ ,hence $\displaystyle \text{ind}_r1=p-1$

Now, you should know that(theorem):

$\displaystyle p-1=\text{ind}_r(-1)(-1)\equiv \text{ind}_r(-1)+\text{ind}_r(-1)=2\text{ind}_r(-1)(\text{mod}(p-1))$.

Some better hints for 2:

Prove:
Let $\displaystyle Q = (2p1, p2....pn)^4 + 1$. Show that Q must have an odd prime factor different than$\displaystyle p1, p2, ...., pn$, and must be of the form 8k + 1.)

Using this:

The odd prime divisors of $\displaystyle n^4+1$ are from the form of $\displaystyle 8k+1$