Show that for every prime p ($p&gt;5$) there exist integers $a$ and $b$ with $1\leq a,b\leq p-1$ such that:  $$\left(\frac{a}{p}\right)=\left(\frac{a+1}{p}\right)=1$$  I tried assuming that there was no integers such that the condition above was true, but I couldn't get to any contradiction.  thank you guys for any hint or adivice you could give me.
For any prime $p\gt 5$,prove that there are consecutive quadratic residues of $p$ and consecutive non-residues as well(excluding $0$).I know that there are equal number of quadratic residues and non-residues(if we exclude $0$), so if there are two consecutive quadratic residues, then certainly there are two consecutive non-residues,therefore, effectively i am seeking proof only for existence of consecutive quadratic residues. Thanks in advance.
Windows is unable to detect disk partitions I installed Windows 8.1 Preview using the DVD yesterday after wiping off my HDD (I made a backup prior to that). While installing Windows I created only the partitions necessary to install Windows (System Reserved, C: shown in second screenshot). Further I installed Ubuntu 13.04. During Ubuntu installation I created necessary extended partitions (for OS, swap and home directory - all ext4 except swap). Later within Ubuntu I created a NTFS partition at the end (partition 4 in first screenshot). This was a primary partition. The objective behind this was that the partition must be accessible from Windows.  While Ubuntu disk utility reports the partitioning just fine (as below)  Windows disk management is showing all partitions as primary.   Thus within Windows I am able to access only C:! What puzzles me is one can't create so many primary partitions! How can I solve this issue? Hyperref: Changing the linkcolor locally in the TOC I found that the link-color for TOC text is identical to the link-color for Figures and Tables. When I change the color to black in the TOC also the color for Figures/Tables citation etc. is changed.  Is there a way to separately set the colors inside the TOC and outside? I recently got a new router, a DRL-600L Router from D-Link, and was setting up some filters to block out certain websites.  One problem I am having is blocking out addresses that use https instead of http.  For example, I'm trying to block out Facebook, because Facebook.   So far, the filter filters http://www.facebook.com perfectly fine, but for the url https://www.facebook.com, it won't block that.  How can I block those kinds of websites? Bootstrap Paradox in Interstellar Connectedness of the Hausdorff distance. Does anyone know a proof of connectedness of the Hausdorff distance?  I mean a proof of the following:  Theorem  If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected.  Remark:  $F(X) = \{ A \subseteq X | A \text{ is nonempty, closed and bounded } \} $, and $d_h$ stands for the Hausdorff distance. Should I use "got" or "gotten" in the following sentence? Installing Office 2010 on Ubuntu 15.04 using Wine I had installed Office 2010 32bit on Ubuntu 12.04 before using guides such as . However, due to package changes, I was not able to replicate the same on Ubuntu 15.04.  How does one install Office 10 using Wine? Where is the "Manipulate Centers Only" button in Blender 2.8? I have looked just about everywhere on the Internet and I can't find the answer to this question. Where the hell is the "Manipulate Centers Only" button in Blender 2.8???? It's the the double arrow icon here:    I seem to have enabled it somehow and I can't scale or rotate my object. Find the number of solutions to $ \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \ldots + \lfloor 32x \rfloor =12345$ Find the number of solutions of the equation    $$\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 8x \rfloor + \lfloor 16x \rfloor + \lfloor 32x \rfloor =12345,$$   where $\lfloor\,\cdot\,\rfloor$ represents the floor function.   My work: I use the fact that $$\lfloor nx \rfloor =\sum_{k=0}^{n-1} \left\lfloor x +\frac kn \right\rfloor.$$ So the equation becomes $$\lfloor x \rfloor +\sum_{k=0}^{1} \left\lfloor x +\frac k2 \right\rfloor +\sum_{k=0}^{3} \left\lfloor x +\frac k4 \right\rfloor +\sum_{k=0}^{7} \left\lfloor x +\frac k8 \right\rfloor   \\ \qquad {}+\sum_{k=0}^{15} \left\lfloor x +\frac k{16} \right\rfloor +\sum_{k=0}^{31} \left\lfloor x+\frac k{32} \right\rfloor   \\ = \lfloor x \rfloor + \left\lfloor x+\frac 12 \right\rfloor + \left\lfloor x+\frac 64 \right\rfloor + \left\lfloor x+\frac{28}{8} \right\rfloor   \\ \qquad {}+ \left\lfloor x+\frac{120}{16} \right\rfloor + \left\lfloor x+\frac{496}{32} \right\rfloor$$  What should I do next? I am using longtable and I cannot find a way to adjust its column spacing. I want a table like this:    But I am getting this:    This is what I am doing for defining table size:   \begin{longtable}{p{.1\textwidth} p{.2\textwidth} p{.2\textwidth}p{.3\textwidth}}     xxxxx &amp; xxxxx &amp; xxx &amp; xxxxx\\     \end{longtable} I'm looking for the method by which the partial derangement formula $D_{n,k}$ was derived. I can determine the values for small values of N empirically, but how the general case formula arose still alludes me.  Any links/books or an explanation will be appreciated.  The formula is:  $D_{n,k} = {n \choose k}!(n-k)$  Links: What is the most preferred and easiest way to do pagination in ASP.NET MVC?  I.e. what is the easiest way to break up a list into several browsable pages.  As an example lets say I get a list of elements from a database/gateway/repository like this:  public ActionResult ListMyItems() {     List&lt;Item&gt; list = ItemDB.GetListOfItems();     ViewData["ItemList"] = list;      return View(); }   For simplicity's sake I'd like to specify just a page number for my action as parameter. Like this:  public ActionResult ListMyItems(int page) {    //... } Do I need a RHEL subscription to install packages? Firefox 37.0.2 not opening (removing firefox did not help) runing ubuntu 14.04 LTS Vector bundle on a quadric $Q$ I'm a bit confused about the concept of ground, and perhaps voltage as well, particularly when trying to analyze a circuit. When I learned about Ohm's law in grade school, I learned how to apply the law to calculate current, voltage, and resistance of simple circuits.   For instance, if we were given the following circuit:    We would be could be asked to calculate the current passing through the circuit. At the time, I'd simply compute (based on the rules given) 1.5V/1Ohms=1.5A.  Later on, however, I learned that the reason the voltage of the resistor would be 1.5V is because voltage is really the difference in potential between two points, and that the difference of the voltage across the battery would be the same as that of the resistor (correct me if I'm mistaken), or 1.5V. I got confused, however, after the introduction of the concept of ground.   The first time I tried to do the current calculation for a circuit similar to the previous circuit on a simulator, the program complained about not having a ground and "floating voltage sources". After a bit of searching, I learned that circuits need ground as a reference point or for safety reasons. It was mentioned in one explanation that one can pick any node for ground, although it's customary to design circuits so there is a "easy place" to pick ground.   Thus for this circuit    I picked ground at the bottom, but would it be okay to pick ground between the 7 ohm and 2 ohm resistor - or any other place? And what would be the difference when analyzing the circuit?  I've read that there are 3 typical ground symbols with different meanings - chassis ground, earth ground, and signal ground. A lot of circuits I've seen used in exercises either use earth ground or signal ground. What purpose is there in using earth ground? What is the signal ground connected to?  Another question: since the ground is at unknown potential, wouldn't there be current flowing to or from ground to the circuit? From what I've read we treat the ground as 0V, but wouldn't there be some sort of effect because of a difference in potential of the circuit and ground? Would the effect be different depending on what ground was used?  Finally: In nodal analysis, one customarily picks a ground at the negative terminal of the battery. However, when there are multiple voltage sources, some of them are "floating". What meaning does the voltage of a floating voltage source have? What are the basics needed to learn quantum computing? Heat production of magnifying glass Why Magnifying glass(convex lens) produces heat when placed in front of sun? When should you use WP_Query vs query_posts() vs get_posts()? It seems like half the tutorials in the  and around the blogosphere use  and half use . They all do similar things, so when should I use one over the others? How do I enable or disable Apport? The system crash dialog is annoying me, how can I turn it off? I'd also like to know how to turn it back on just in case I need it to report a problem.