Proving 3 as a Quadratic Non-Residue of Mersenne Primes | Number Theory Problem

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To prove that 3 is a quadratic non-residue of all Mersenne primes greater than 3, one can utilize the law of quadratic reciprocity. Starting with small Mersenne primes, such as 7 and 31, helps identify patterns that support this claim. The discussion emphasizes the importance of working through examples to build understanding. Participants suggest that examining specific cases can clarify the concept. Engaging with the law of quadratic reciprocity is crucial for solving the problem effectively.
ak_89
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I was just working on some problems from a textbook I own (for fun).
I am not sure how to start this problem at all.

Here's the question: Show that 3 is a quadratic non-residue of all Mersenne primes greater than 3.

I honestly don't know how to start. If I could get some help to push me in the right direction that would be great.

Thanks
 
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Do you know the law of quadratic reciprocity? If so, use it to work out examples for small Mersenne primes and look for some patterns.

Petek
 
That helped me a lot. Thanks a bunch :wink:
 

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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