$$\left(\frac{1\cdot2}{p}\right) + \left(\frac{2\cdot3}{p}\right) + \left(\frac{3\cdot4}{p}\right) +\ldots+ \left(\frac{(p-2)(p-1)}{p}\right) = -1$$  Note: $\left(\frac{a}{b}\right)$ represents the Legendre Symbol.  I have tried using this method. For each $k$ between $1$ and $p-2$ denote by $k'$ its multiplicative inverse $\mod p$. To estimate the sum of all Legendre symbols $\left(\frac{k(k+1)}p\right)$ show first that $\left(\frac{k(k+1)}p\right) = \left(\frac{1+k'}p\right),$ then estimate the sum of $\left(\frac{1+k'}p\right).$
Let $p$ be an odd prime number. Prove that $$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \left( \frac{3 \cdot 4}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$$ where $\left( \frac{a}{p}\right)$ is the Legendre symbol.  This seems to be a tricky one! I've tried using the property $\left( \frac{ab}{p} \right)=\left( \frac{a}{p}\right) \left( \frac{b}{p} \right)$ and prime factoring all the non-primes but to no avail. I had a quick thought of induction haha, but that was silly. I tried factoring the common Legendre symbols like $\left( \frac{3}{p}\right) \left[ \left( \frac{2}{p} \right) + \left( \frac{4}{p} \right) \right]$ but that didn't bring anything either. And I've been looking for pairwise cancellation with $1$ term leftover, but that does not seem to work.  Can you help?
What are the differences between $$, \[, align, equation and displaymath? Discrepancy with lsof command when trying to get the count of open files per process Would this look natural:     StackExchange is a helluva site! I found out about this site at Facebook.    Or should we use the preposition on:     StackExchange is a helluva site! I found out about this site on Facebook.    Or are both okay?   (Inspired by a ) Schengen visa in US without H1B stamp Prevent parameter expansion in shell script I often have to do a recursive search in files. Tired of typing the whole "find/grep" combination all the time, I just created a script with the following line:  find . -name $1 -exec grep $2 {} + 2&gt;/dev/null   The idea is that I should be able to run it as, for example:  myfind '*' hello   The problem is, when '*' gets passed into the script, the script is expanding it to each file in the directory.  I tried putting a quote around $1:   find . -name '$1' -exec grep $2 {} + 2&gt;/dev/null   However, this turns $1 into a literal string.  Would appreciate your help in finding the right syntax. Must be something simple that I am missing. Seeing through objects in Game engine function $f(x)=x^{-(1/3)}$ , please check whether my solution is correct? Let $f(x)=x^{-(1/3)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from -1 to 1. Which of the following statements is/are TRUE?   f is continuous in [-1, 1] f is not bounded in [-1, 1] A is nonzero and finite     I try to explain :    graph is  :     False , since left limit is not equivalent to right limit. True , since  f(x) rises to $infinite$ at $x=0$. True , since we can calculate it by doing integrating the function Why crontab scripts are not working? Often, crontab scripts are not executed on schedule or as expected. There are numerous reasons for that:   wrong crontab notation permissions problem environment variables   This community wiki aims to aggregate the top reasons for crontab scripts not being executed as expected. Write each reason in a separate answer.  Please include one reason per answer - details about why it's not executed - and fix(es) for that one reason.  Please write only cron-specific issues, e.g. commands that execute as expected from the shell but execute erroneously by cron. Keytool Signing Problem: Keystore was tampered with, or password was incorrect systemd gives us the systemctl commands suite which is mostly used to enable services to start at boot time. We can also start, stop, reload, restart and check status of services with the help of systemctl.  We can do, for example, sudo systemctl enable service_name, and service_name will automatically start at boot time. We can also disable services not to start at boot time.  Is the only difference between the service and systemctl commands that systemctl can be used to enable the start of services at run time? Can we use systemctl on any service? What other significant differences are there? Best way to implement keyboard shortcuts in a Windows Forms application? I'm looking for a best way to implement common Windows keyboard shortcuts (for example Ctrl+F, Ctrl+N) in my  application in C#.  The application has a main form which hosts many child forms (one at a time). When a user hits Ctrl+F, I'd like to show a custom search form. The search form would depend on the current open child form in the application.  I was thinking of using something like this in the ChildForm_KeyDown event:     if (e.KeyCode == Keys.F &amp;&amp; Control.ModifierKeys == Keys.Control)         // Show search form   But this doesn't work. The event doesn't even fire when you press a key. What is the solution? Let   $$f(x) = (x-1)(x-2)(x-3)(x-4)(x-5),\quad -\infty&lt; x&lt;\infty.$$   The number of distinct roots of equation   $$\frac{d}{dx}f(x) = 0$$   is exactly ?     The only method that I know is to multiply and then find the derivative of function then apply Sturm's theorem but it seems vague when you have to solve the question in 3 to 5 minutes . So you are requested to suggest a plausible alternative approach. RECENTLY ON THE ANDROID SITE WE GOT  WRITTEN IN ALL CAPS.  THIS WAS SURPRISING TO ME, SINCE I ASSUMED IT WOULD HAVE BEEN CAUGHT BY THE QUALITY FILTER &mdash; BESIDES BEING INDICATIVE OF OTHER POST QUALITY ISSUES, IT'S AWFUL TO LOOK AT AND REQUIRES A LARGE AMOUNT OF EDITING TO FIX (EVEN IF YOU RUN IT THROUGH TOLOWER() FIRST).  WE ALL KNOW THAT CAPS LOCK IS , BUT STILL: CAN WE BLOCK THESE QUESTIONS?    RICK  RICK  OH MAN THAT STUFF JACOB G  IS COMEDY GOLD  RIIICK     Note to editors: This post looks exactly like it is supposed to look, as an example of a question in all-caps. Please do not replace the uppercase letters with lowercase. Resources for Blender Since asking for tutorials, videos, resources, links to other content or downloads is considered off topic here, this is a specifically created Community Wiki which gathers resources for Blender and it has been approved by the .  Just write in the appropriate answer/section. If you have concerns, questions, post a  question, so we don't clutter the comments, but you can link your meta question from the comments.  Questions regarding such resources are not allowed anymore, except for very specific and on topic requests (ask on  if you're unsure about your question). Follow the instructions made in the question about how to post, what can be posted, etc.  Learning             Assets    (also see: )   Reference Images/Blueprints (also see: )    Tools    (also see: )      Other   Special Interest ()     Also see the . Limiting sessions to one IP at a time On a site with premium user account subscriptions, I'd like to be able to limit logins to one computer at a time. The most straightforward way to accomplish this would be to limit by IP, but I haven't had any luck finding a plugin to accomplish this. Does anybody know of one I can use to get this functionality?  Related: I am trying to rename my Windows / Boot Camp partition from "EFI Boot" to "Boot Camp" or "Windows". But I do not figure out where to rename.   I tried bless command on the NTFS Volume, I tried creating the folders "System/Library/CoreServices" and copied the .disklabel file into it (or also blessit) but nothing changed.  Finally I got the Volume Icon on Boot Selection to work by copying the icns File onto the EFI partition. Also tried blesson the EFI partition – but this also do not work.  So I have my Icon, but I also would like to rename it – how can I do this?  Systems are Windows 10 and macOS Mojave. Can not add image inside SVG via jQuery:  tag becomes This is a follow up question to my previous question  Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the extension of $I(V)k[x_1,\ldots,x_{m+n}]$. Now I believe I have proven this (see the proof at the bottom of my question) but when I look at Martin's answer , it is instead claimed that we  actually have  $$I(V \times W )= \sqrt{I(V) + I(W)},$$  and for $I(V) + I(W)$ to be a radical ideal we need $k$ to be algebraically closed.           My question is: What's going on here? I believe my claim is true even without the assumption that $k$ is algebraically closed.           Here's a proof user Sanchez told me of, which I have simplified:       First it is clear that we always have $I(V) + I(W) \subseteq I(V \times W)$. For the reverse inclusion consider a polynomial $f \in I(V \times W)$. Then we can always write $$f = \sum_{i=1}^n f_ig_i$$ where $f_i \in k[x_1,\ldots,x_n]$ and $g_i \in k[x_{n+1},\ldots,x_{m+n}]$. Now take any $b' \in W$. If for all $i$ we have $g_i(b') = 0$ then since $b'$ is arbitrary, $g_i \in I(W)$ for all $i$. Then $f \in I(W) \subseteq I(V ) + I(w)$ and we are done. Otherwise suppose there exists $b \in W$  and $i$ such that $g_i(b) \neq 0$. Then wlog we may suppose that  $g_1(b) \neq 0$.   Next, $\sum f_ig_i(b) = 0 $ on all of $V$ by assumption of $f \in I(V \times W)$. So $\sum f_i g_i(b) = p$ for some $p \in I(V)$. Now write $$f_1 = \frac{ p - g_2(b) f_2 + \ldots g_n(b)f_n}{g_1(b)}.$$  Substituting this for $f_1$ in $\sum f_ig_i$, followed by taking things mod $I(V)$ we get an  expression with only $n-1$ terms $\mod{I(V)}$. Continuing this process we will finally get an expression with $0$ terms $\mod{I(V)}$ so that $f \in I(V)$. This shows $I(V \times W) \subseteq I(V) + I(W)$ which completes the proof. Which site deals with career development questions for programmers? Is there any site in the network where career development questions about the programming profession can be asked ? I thought Programmer's was the site for it, but that doesn't appear to be the case. Carrying confidential corporate data into US on laptop/Drive/device - how to prevent data theft? If I travel to the US and had privacy concerns (for instance may be I was afraid that a rogue customs official may steal said data) what is the current recommended best practice to avoid this?