Timeline for A simple number theory confirmation
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 21, 2019 at 7:37 | comment | added | Turbo | @js21 Your proof seems to show only $ \binom{\alpha}{\beta}$ and two of $$\binom{-\beta}{\alpha},\binom{\alpha+\beta}{\alpha -\beta},\binom{\beta -\alpha}{\alpha +\beta} $$ suffices to form an integral basis of $\mathbb Z^2$. So perhaps the right problem should perhaps be 'if three of $$x_1a+x_2b,\mbox{ }x_2a-x_1b,\mbox{ }x_1\frac{(a+b)}2+x_2\frac{(a-b)}2,\mbox{ }x_2\frac{(a+b)}2-x_1\frac{(a-b)}2$$ are in $\Bbb Z$ for some $x_1,x_2\in\Bbb R$ then $x_1,x_2\in\Bbb Z$ should hold?'? | |
| Apr 17, 2017 at 17:03 | vote | accept | Turbo | ||
| Apr 17, 2017 at 17:00 | comment | added | David Cohen | @Turbo, We know that $x_1y_1+x_2y_2$ is an integer whenever $(y_1,y_2)$ is in the group generated by the four vectors. js21 shows that this group includes the vectors $(1,0)$ and $(0,1)$, which implies directly that $x_1=1\cdot x_1+0\cdot x_2$ and $x_2=0\cdot x_1+1\cdot x_2$ are integers. | |
| Apr 17, 2017 at 15:57 | comment | added | Turbo | I am not seeing directly since I am not familiar with lattice terminology - why does this show '$x_1,x_2\in\Bbb Z$ has to hold' ? | |
| Apr 17, 2017 at 9:43 | history | answered | js21 | CC BY-SA 3.0 |