Number Theory Questions: Proving p and x2 Congruencies

[ak_89]

I have a few questions I am having troubles with. If someone can push me in the right direction that would be awesome. Here are the questions:

1. Prove that the prime divisors, p cannot equal 3, of the integer n²-n+1 have the form 6k+1. (Hint: turn this into a statement about (-3/p) )

2. Show that if p is congruent to 1 (mod 4), then x² is congruent to -1 (mod p) has a solution given by the least residue (mod p) of ( (p-1)/2)!

I honestly have no idea how to start. I would greatly appreciate some help.
Thanks

 

[Yoshara]

For the first question, try multiplying the n²-n+1 term by some integer and rearranging things...

Unfortunately, I'm stumped on how to do the second question

 

[ak_89]

Thanks! I got that proof. But I am still stuck on the second question as well. I played around with it.. but I have yet to get anywhere that is useful to prove the question.

I could really use some help.

 

[Petek]

Hint for #2: Use Wilson's Theorem.

Petek