My question is simple: If all integers can be represented by $2k$ or $(2k+1)$ where all odd numbers have the form $(2k+1)$ the square of any odd number should be in the form of $(4k^2 + 4k +1) = 4q+1$ then why we have to further represent $q$ (in the last equation) as $2k$ and $2k+1$ to make a conclusion that odd number squared is always is in the form of $8q+1$ ? didn't we already assumed that any odd number is in the form $2k+1$ where $k$ can be any integer.  if that isn't the case then why should not we further replace $q$ as $2k$ and $2k+1$ and further and this wont stop. so what is logic here? The proof makes sense if we start with odd integer as $4k+1$ or $4k+3$ but again how we decide which form should we start with, because both (2k and 4k version) represent the odd numbers.
I am trying to show that if $n$ is an odd natural number, then $8$ divides $n^{2}-1.$ I was able to prove that because I know that if $n$ is an odd natural number, then $n^{2}$ can be written as $8k+1$ for some $k\in \mathbb{N}.$ I would like to show this question by using the Euclidian division. Then I wrote $n^{2}-1=8k+r$, where $0\leq r  &lt; 8.$ When $r=2$ we get $n^{2}-1=8k+2.$ Since $n$ is odd, then $n^{2}-1$ is even and I got stuck. Is there way to fix that?
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I cast  on my opponent, whose hand has two Mountain cards. Wistful Thinking resolves and my opponent draws a  and another Mountain.  Can my opponent cast Shock before discarding it? I've just upgraded from Ubuntu 18.04 to its newest LTS version - 20.04. While it comes with some interesting changes, the Desktop icon system was messed up in my opinion. When you hover the mouse cursor over the icons, you see a huge space marked around them that cannot be violated (as you can see in the image below) so that they need to be placed very far apart from each other.  18.04 allowed you to place icons much closer to each other and didn't have that marked space around them. This new way leaves much less space on the Desktop to organize large amounts of icons - including folders, files and shortcuts. Is there a way I can significantly reduce the size of icons and their "spaces" or go back to the old system of organizing files? 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In wikipedia I can see an equation that seems to give all the conserved quantities () but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain. $\displaystyle \int\nolimits_0^8 \sqrt{x^4+4x^2}\,dx$.   Alright, so I thought I had this figured out. Here's what I did:   I factor out an $x^2$ to get $\sqrt{x^2(x^2+4)}$. I let $x = 2\tan(\theta)$, therefore the integrand is $\sqrt{4\tan^2(\theta)  (4\tan^2(\theta) + 4)}$. Factor out a 4 and it becomes $\sqrt{(16\tan^2(\theta)  (\tan^2(\theta) + 1))}$ Which equals $\sqrt{16\tan^2(\theta) \sec^2(\theta)}$ This is easy to take the sqrt of. The integrand becomes $4\tan(\theta)\sec(\theta)$. Now, the integral of this is $4\sec(\theta)$ And it's evaluated from $0$ to $\arctan(4)$ right? Because as $x$ goes to $0$, so does $\theta$, and as $x$ goes to $8$, $\theta$ goes to $\arctan(4)$... 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