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

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 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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