Number Theory Questions: Proving p and x2 Congruencies

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The discussion focuses on two number theory questions regarding congruences and prime divisors. The first question asks for a proof that the prime divisors of the integer n² - n + 1, excluding 3, take the form 6k + 1, with a hint to relate it to the Legendre symbol (-3/p). The second question seeks to demonstrate that if a prime p is congruent to 1 modulo 4, then the congruence x² ≡ -1 (mod p) has a solution, with a suggestion to apply Wilson's Theorem. Participants express their struggles with both proofs, seeking guidance and hints to progress. The conversation emphasizes the importance of foundational concepts in number theory for tackling these problems.
ak_89
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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
 
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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 :rolleyes:
 
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.
 
Hint for #2: Use Wilson's Theorem.

Petek
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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