Given $p$ a prime number such that $p \geq 5$  Show that $p^2$ is congruate to $1$ modulo $24$  This is what I tried :  We have $p^2$ congruate to 1 modulo $3$ because if $p=3k+2$ so p is congruate to -1 modulo 3 so p^2 is congruate to 1 modulo 3  And $p^2$ is congruate to $1$ modulo $8$ because $p=2k+1$ ($p\geq5$) so $p^2-1=4q(q+1)=8q'$  We have $p^2=3k+1$ and $p^2=8k'+1$  So $p^4=24kk'+3k+8k'$ and $3k=p^2-1$ and $8k'=p^2-1$ So :  $p^4-2p^2+2=24kk'$  Im stuck here ! I don't want use to use a proprietie directly
Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
How do I delete/remove a shell function? I have done this:  $ z() { echo 'hello world'; }   How do I get rid of it? Complete classification of the groups for which converse of Lagrange's Theorem holds I wish to show that $\mathbb{Q}(\sqrt2, \sqrt3)$ is not isomorphic to $\mathbb{Q}(\sqrt[4]2)$.    What I did so far was to show that $\mathbb{Q}(\sqrt2,\sqrt3) =\text{span}\{1,\sqrt2,\sqrt3,\sqrt6\}$.  Assuming they are isomorphic, denoting $f$ the isomorphism, then $f(1) = 1$, so $x^2-3$ has a root in $\mathbb{Q}(\sqrt2,\sqrt3)$ so $\sqrt 3\in\mathbb{Q}(\sqrt[4]2)$. ($\sqrt2 \in \mathbb{Q}(\sqrt[4]2)$, $\sqrt[4]2\cdot\sqrt[4]2=\sqrt2)$.   Thus under the assumption, $\mathbb{Q}(\sqrt2, \sqrt3)\subseteq \mathbb{Q}(\sqrt[4]2)$. As a vector spaces of the same dimension over $\mathbb{Q}$, it means that $\mathbb{Q}(\sqrt2, \sqrt3)= \mathbb{Q}(\sqrt[4]2)$.  Now all is left to do it to show a contradiction: $\sqrt[4]2\in\mathbb{Q}(\sqrt2,\sqrt3)$, then $\sqrt[4]2 = a+b\sqrt2+c\sqrt3+d\sqrt6$ so $2=(a+b\sqrt2+c\sqrt3+d\sqrt6)^4$, I believe that opening the expression, I would find out that it can't be an element of $\mathbb{Z}$. However this is a really inelegant way (if it really works), and I wish for a better way to do so.  An other approach: $\mathbb{Q}(\sqrt2,\sqrt3) = \mathbb{Q}(\sqrt2 + \sqrt3)$ by the irreducibility of $x^4-10x^2+1$ over $\mathbb{Q}$ (an some dimension theorem usage). Also, $x^4-10x^2+1$ splits in $\mathbb{Q(\sqrt2, \sqrt3)}$.   Is there a way to show that this polynomial doesn't split over $\mathbb{Q}(\sqrt[2]4)$? with different method then assuming that $\sqrt3\in\mathbb{Q}(\sqrt[4]2)$ and getting a contradiction? Maybe the fact that $\sqrt[4]2$ has complex roots might help? How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them. Factoring $-3x^2 - 2xy + 8y^2 + 11x - 8y - 6$ Why does a vertical line have a SLOPE (steepness) of undefined? The curve shows :   x goes 1 to right  y goes up by 3   the SLOPE is defined as the ratio of rise to the step  SLOPE = rise/step = 3/1 = 3  Now that :   Horizontal line has SLOPE of 0. Vertical line has SLOPE of undefined.   It makes sense for me to imagine horizontal line has SLOPE of 0 since there is no rise at all. But why the vertical has SLOPE of undefined.? Should we close 'fix my program' questions Recently, I've seen many questions with this structure:   a wall of code a request for the community to debug it by inspection   You can find a nice selection .  I offer, as a thesis, that these questions are not consistent with the stated goals of SO. They do not, usually, create a resource of use to other people.   As a community, we could politely but firmly tell these askers to do enough of their own work to come up with a concrete, specific, programming question, other than 'can someone tell me what's wrong with this?'  Don't get me wrong. I'm sympathetic to people with problems. I have not been downvoting or voting to close these, because the community seems to be inclined in general to be helpful. I'm asking, are we really doing the right thing by encouraging the site to populate with these?  If the consensus is 'yes' then I'll just slink back to my cave. I'm not interested in starting a campaign here. How can I sum two values and store the result in other variable? UK Train - Can I board at a different station Parenthesis vs brackets for matrices When I first learned linear algebra, both the professor and the book used brackets like [ and ] to enclose matrices. However, in my current differential equations textbook, matrices are enclosed by parenthesis, and I suddenly realize everybody else are using them too.  So are brackets/parenthesis for enclosing matrices always totally interchangeable? Let $(a_k)$ be a sequence of real numbers and let $b_k=\frac{a_1+a_2+\dots a_k}{k}$ for each $k\in \mathbb{N}$.  Prove that if $(a_k)$ converges to $\alpha\in \mathbb{R}$, then the sequence $(b_k)$ also converges to $\alpha$.  My answer: Let $\epsilon &gt; 0$. Since ($a_k$) converges to $\alpha$, then there exists a $N_{\epsilon1}\in\mathbb{N}$ such that $|a_k - \alpha | &lt; \frac{\epsilon}{2}$ for all $k \geq N_{\epsilon1}$. Notice that $\frac{a_1+...+a_n - n\alpha}{k}$ converges to $0$ as $k$ approaches infinity. Thus, there exists $N_{\epsilon2}\in\mathbb{N}$ such that $|\frac{a_1+...+a_n - n\alpha}{k}|&lt;\frac{\epsilon}{2}$ for all $k&gt;N_{\epsilon2}$. Now, let $N= $max$ (N_{\epsilon1},N_{\epsilon2})$. Then for any $k&gt;N$ we get,  $$\begin{array}{rcll} |b_k-\alpha| &amp;=&amp; |b_k - \alpha*1|&amp;\\ &amp;=&amp; |\frac{a_1+...+a_k}{k} - \frac{k\alpha}{k}|&amp;\\ &amp;=&amp; |\frac{a_1+...+a_n}{k} + \frac{[a(n+1)+...+a_k]}{k} - \frac{(k-n)\alpha}{k} - \frac{n\alpha}{k}|&amp; \text{(notice that $N_{\epsilon1}\leq N&lt;k$)}\\ &amp;=&amp; |\frac{a_1+...+a_n - n\alpha}{k} + \frac{[a(n+1)+...+a_k - (k-n)\alpha]}{k}|&amp;\\ &amp;=&amp; |\frac{a_1+...+a_n - n\alpha}{k} + \frac{[(a(n+1) -\alpha) + ... + (a_k - \alpha)]}{k}|&amp;\\ &amp;=&amp; |\frac{a_1+...+a_n - n\alpha}{k} + \frac{(a(n+1) -\alpha)}{k} + ... + \frac{(a_k - \alpha)}{k}|&amp;\\ &amp;\leq&amp; |\frac{a_1+...+a_n - n\alpha}{k}| + |\frac{(a(n+1) - \alpha)}{k}| + ... + |\frac{(a_k - \alpha)}{k}|&amp; \text{(by the triangle inequality)}\\ &amp;=&amp;|\frac{a_1+...+a_n - n\alpha}{k}| + \frac{|(n+1) - \alpha|}{k} + ... + \frac{|a_k - \alpha|}{k}&amp;\\ &amp;&lt;&amp; \frac{\epsilon}{2} + [\frac{\epsilon}{2}/k + ... + \frac{\epsilon}{2}/k]&amp; \\ &amp;=&amp; \frac{\epsilon}{2} + \frac{\epsilon}{2}*\frac{(k-n)}{k}&amp;\\ &amp;&lt;&amp; \frac{\epsilon}{2} + \frac{\epsilon}{2}&amp; \text{(since $k-n&lt;k$)}\\ &amp;=&amp; \epsilon&amp; \end{array}$$ And since we found such an $N$ for arbitrary $\alpha$, then $(b_k)$ converges to $\alpha$ by definition.  I'm not sure i kept all my variables straightened out. How does this look and if you have a better idea let me know! Why is 128==128 false but 127==127 is true when comparing Integer wrappers in Java? I'm trying to connect to a sandbox using the sfdx cli and the JWT flow through force:auth:jwt:grant. According to , we can use the same connected app information from Production to connect to our Scratch Org.  Is it something that should work for sandbox too ? I'm getting a "Failed: Not approved" error.  Connected app in sandbox comes from the refresh but has a different consumer key. The Permission set is set correctly on my user.  Is there a way to connect to a recently refreshed sandbox using the JWT flow (without manually changing something like the consumer key) ? Is it acceptable to change the vid column of a term programmatically in the database? I am totally new to Linux, servers and websites. I recently set up Apache2, Python, PHP, and MySQL for setting up a LAMP server. I am using Xubuntu 16.04 and PHP 7.0.11. I followed a guide and placed a PHP file named testing.php in it with:  &lt;?php phpinfo(); ?&gt;    It doesn't show anything on localhost/testing.php All it shows is the code itself. I'm consistently receiving significant delays on Stack Overflow when opening the flag dialog on questions. Delays do not occur on any other StackExchange site I have checked. This started a couple days ago (as far as I can tell).  Delays occur not only when loading the initial screen, but also when loading the submenus. This is slightly confusing for a couple seconds (until you're used to it) since it appears as if the flag dialog has simply vanished.   What's going on here? I demand to know, otherwise somebody will be very upset.  Not me. I'm not going to get that upset over something like this.  But someone will.    Specs (though I know of several others who have this issue):  Windows 7 64-bit Chrome Version 30.0.1599.69m     Ping: 31ms Download: 6.88mbps   Upload: 1.07mbps Why do all the planets of the solar system orbit in roughly the same 2D plane? Most images you see of the solar system are 2D and all planets orbit in the same plane. In a 3D view, are really all planets orbiting in similar planes? Is there a reason for this? I'd expect that the orbits distributed all around the sun, in 3D. Has an object made by man (a probe) ever left the Solar System? 1099 for settlement what about lawyer fees? I am getting a settlement from a business over a suit but most of the settlement will go to the attorney to cover their fees incurred on contingency.  If the 1099 reports the full settlement, will I have to pay taxes on the full amount or if not, how do I tell the IRS it is not all taxable. Otherwise, I could easily end up owing more in taxes than I get for my small portion. How to do relative imports in Python? How can I fix a 404 Error when using a PPA or updating my package lists? PPA seems to be constantly offline. Whenever I use sudo apt-get update, this error is shown:  W: Failed to fetch http://ppa.launchpad.net/ 404  Not Found   How do I fix these errors?