Thread: Index Arithmetic Questions

1) Show that if p is an odd prime and r is a primitive root of p, then (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.)

For the first Q.

is a primitive root of ,hence

Now, you should know that(theorem):

.


Some better hints for 2:


Prove:
Let . Show that Q must have an odd prime factor different than , and must be of the form 8k + 1.)

Using this:

The odd prime divisors of are from the form of