Hi, I just registered for this site, and it looks like a pretty neat place.

I just needed a confirmation that I did a couple proofs right. I'm taking Number Theory at my university, and on Tuesday I have to teach the class two proofs.

My first one is to prove that

has exactly two solutions: 1 and p-1

proof: let

be any solution. SO we have

so

thus

Because p is a prime (this is important) . Correct.
otherwise expressed as

. So

. Since r is a least residue (modp) we have r = p -1 or 1

Correct. Note that ...
The other one I have to do is prove that if (a,m)

*doesnt* divide b, then

has no solutions.

Proof: We can prove this by looking at the contrapositive, which is logically equivalent. So if

then

has a solution.

This is not the contrapositive! It is "if has a solution then .

Fix this.

Tonio
Let r be a solution, so

. Thus,

and there exists an integer k such that

. Since

and

it follows that

Thanks guys I appreciate it