Mod 8

For students of class 6-8 (age 12 to 14)
User avatar
Thamim Zahin
Posts:98
Joined:Wed Aug 03, 2016 5:42 pm
Mod 8

Unread post by Thamim Zahin » Thu Jan 12, 2017 9:23 pm

If the $x^2+2y^2$ is an odd prime, then it has the form $8n+1$ or $8n+3$.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

dshasan
Posts:66
Joined:Fri Aug 14, 2015 6:32 pm
Location:Dhaka,Bangladesh

Re: Mod 8

Unread post by dshasan » Thu Jan 12, 2017 10:42 pm

$x$ 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. :D
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

Post Reply