(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]p[/itex] be an odd prime. Show that [itex] x^2 \equiv 2 (mod p)[/itex] has a solution if and only if [itex]p \equiv 1 (mod 8)[/itex] or

[itex]p \equiv -1 (mod 8) [/itex]

3. The attempt at a solution

Ok, I figured the more of these I try, the better I'll get at them. Assuming that

[itex] x^2 \equiv 2 (mod p)[/itex] has a solution first. I get

[tex] 1 = ( \frac{2}{p} )= -1^{\frac{p^2 - 1}{8}} [/tex]

So [tex] 1 = -1^{\frac{p^2 - 1}{8}} [/tex].

This implies that [itex]\frac{p^2 - 1}{8}[/itex] must be even. So

[itex] \frac{p^2 - 1}{8} = 2k[/itex] for an integer [itex]k[/itex].

I'm not sure if I'm on the right track though. Any hints would be appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Another number theory proof

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**