Mod 8
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 pm
If the $x^2+2y^2$ is an odd prime, then it has the form $8n+1$ or $8n+3$.
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Re: Mod 8
$x$ is odd. So, $x^2 \equiv 1(mod 8)$. Now, If $y$ is even, then $2y^2 \equiv 0(mod 8) \Rightarrow x^2 + 2y^2 \equiv 1(mod 8)$. If $y$ is odd, then $y^2 \equiv 1 (mod 8) \Rightarrow 2y^2 \equiv 2 (mod 8) \Rightarrow x^2 + 2y^2 \equiv 3 (mod 8)$. So, $x^2 + 2y^2$ has the form $8n + 1$ when y is even and $8n+3$ when y is odd.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
- Charles Caleb Colton
- Charles Caleb Colton