Is There a Proof for p=1 mod 4 if p|x^2+1?

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Homework Statement



Let n be a whole number of the form ##n=x^2+1## with ##x \in Z##, and p an odd prime that divides n.
Proof: ##p \equiv 1 \mod 4##.

Homework Equations



The Attempt at a Solution



The only relevant case is if p=3 mod 4.

If I try to calculate mod 3, or mod 4, or mod p, I'm not getting anywhere.
 
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Welcome to PF, TDA120! :smile:

Someone gave me a hint: what is the order of x in ##\mathbb{Z}/p\mathbb{Z}^\times##? :wink:

Happy biking!
 

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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