Show that if $n$ is an integer, then every prime factor of $4n^2+1$ is congruent to $1 \pmod 4$. (Hint: if $p\mid 4n^2+1$), then what can you say about $(-1\mid p)$?  Approach:  I went over all the cases for n and concluded that $4n^2+1 \equiv 1\pmod 4$. We know that $4n^2+1$ can be represented as the product of prime numbers, so let $4n^2+1=q_1\cdots q_\ell$. This implies that $q_i \mid 4n^2+1$  b) Show that there are infinitely many primes congruent to 1(mod 4)  Approach  The common way to do this is by contradiction. Assume there are finitely many primes congruent to 1 mod 4. Call them $p_1,....,p_k$. Now let's build a number n congruent to 1 mod 4 with these primes. n=$4p_1*....*p_{k}+1 \equiv 1(mod\text{ } 4)$. By definition we can represent n as a product of primes. Let $n=q_1,...,q_l$. If $q_j=p_j$ then $q_j | 4p_1*...*p_k$ and $q_j | 1$ which can't be possible, so contradiction.  I am just wondering if there is a way to do the latter by using what was just done.
Given an integer $n$, we are asked to investigate about the existence of integer divisors of $n^2+1$ of the form $4k+3$. Can you provide some insights about it?
I try to clean my mid-2015 Macbook Pro with Retina display by wiping the screen with a damp cloth and then immediately drying.  However, there seems to be left some residue/streak marks, which look very oily. Sometimes these disappear after a few minutes, but they are still very unsightly, and a bit worrying (is there something wrong with my screen?)  I'm wondering if there are other cleaning techniques which avoid this issue, or whether this is completely normal behaviour.  Thank you! Automate on-site plagiarism detection There have been many cases where on-site plagiarism has been coincidentally discovered by  good citizens either while looking for , or while answering questions merely to find out one of their  etc.   Most of the cases have been identified as verbatim copy-paste of the original answer.  I understand plagiarism is considered very offensive and can't be tolerated. It is very unprofessional and unfair. So here are my questions (and possibly a feature request):   Could verbatim plagiarized content be automatically detected?  Should it be pushed into a review queue similar to low quality posts for other users to confirm? Is there a way to set an "Adventure Mode Boundary" in Minecraft? Is there a way to set an "Adventure Mode Boundary" in Minecraft?  I am trying to create this village where players can spawn in, and then set out to explore the world and build and do normal minecraft stuff. I don't want them to destroy this village, so I want them to be in adventure mode when they are in the village and then switch them to survival when they leave, and vice-versa. When they return to the village, they get switched back to adventure mode.  In other words, I want to set an "Adventure Mode Boundary" to protect buildings. I am working without mods. I am hoping there is an simple way using command blocks. 50 question per month limit? I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) Computing the $n^{\textrm{th}}$ permutation of bits. I've seen  post about the $n^{\textrm{th}}$ permutation of a set but that is not what I need. If you have a bit string (ones and zeros only) there are  to quickly permute the NEXT lexicographically ordered bit permutation. For example take $$000111 \rightarrow 001011 \rightarrow 001101\rightarrow\cdots$$ etc. If the string is long then there are going to be approximately one Bajillion of these things. And I want to know how to compute the $n^{\textrm{th}}$ guy without doing the exhaust.  Backgound: This is for a parallel computing job where I need to farm out the search space of elements of a set.  Foreground: I own a copy of Knuth's 4th volume of "The Art of Computer Programming" and I think the answer is in there but I can't seem to find it. (It's like 900 pages).  I'm posting here in the hopes that someone has knowledge of this (obviously). Even if you can point me to a source, say the part in Knuth's book where he describes this problem I would be most grateful. Optimal drive configuration for SQL Server 2008R2 I have a fairly busy database server running SQL Server 2008 R2 that has the following setup:   SATA RAID 1 (2 Drives)  - OS / Programs SAS  RAID 10 (4 Drives) - Sql Database Files (data and logs) SAS  RAID 1 (2 Drives)  - TempDB (data and logs)   Assuming I can't add additional drives into this server, have I made the best use of the configuration I have available? Or should I consider another scheme here where logs are isolated from the data files, for example?  Update:  For those that requested further hardware details:   The SATA drives (used for the OS / Program partition) are: WD 7200 RPM 3 Gb/s 3.5 Inch SATA The SAS drives used in the other arrays are: Seagate 15K RPM 6 Gb/s 3.5 inch SAS The RAID controller used is an: LSI 9260-8i SAS/SATA 6 Gb 8 port   Update 2:  Based upon the feedback I've received, it looks like I have the following viable options to choose from - I will award the bounty to someone that can tell me which is likely to be the best in the environment that I've outlined:   Leave everything as is - I probably won't do much better  Move my 2 SAS RAID 1 drives into my existing RAID 10 array to have it composed of 6 disks in total  Move my log files onto the SAS RAID 1 and/or relocate TempDB (data or logs) back to the RAID 10 How can I require an optional Perl module if installed? I have Perl code which relies on Term::ReadKey to get the terminal width. My installation is missing this module, so I want to provide a default if the module isn't present rather than throw an exception. How can I conditionally use an optional module, without knowing ahead of time whether it is available. # but only if the module is installed and exists use Term::ReadKey; ...  How can I accomplish this? line 56 syntax error: unexpected end of file I have typed out the code for this mychoice.sh bash example and I keep getting the syntax error: unexpected end of file.  Am I missing something?  mychoice.sh  #!/bin/bash  E='echo -e';e='echo -en';trap ";exit" 2 ESC=$( $e "\e") TPUT(){ $e "\e[${1};${2}H";} CLEAR(){ $e "\ec";} CIVIS(){ $e "\e[25l"} DRAW(){ $e "\e%\e(0";} WRITE(){ $e "\e(B";} MARK(){ $e "\e[7m";} UNMARK(){ $e "\e[27m";} R(){ CLEAR ;stty sane;$e "\ec\e[37;44m[J";}; HEAD(){ DRAW     for each in $(seq 1 13);do     $E "   x                    x"     done     WRITE;MARK;TPUT; 1 5     $E "BASH SELECTION MENU         ";UNMARK;}     i=0; CLEAR; CIVIS;NULL=/dev/null FOOT(){ MARK;TPUT 13 5     printf "ENTER - SELECT,NEXT         ";UNMARK;} ARROW(){ read -s -n3 key 2&gt;/dev/null &gt;&amp;2      if [[$key = $ESC[A ]];then echo up;fi      if [[$key = $ESC[B ]];then echo dn;fi;} M0(){ TPUT  4 20; $e "Login info";} M1(){ TPUT  5 20; $e "Network";} M2(){ TPUT  6 20; $e "Disk";} M3(){ TPUT  7 20; $e "Routing";} M4(){ TPUT  8 20; $e "Time";} M5(){ TPUT  9 20; $e "About  ";} M6(){ TPUT 10 20; $e "Exit   ";} LM=6 MENU(){ for each in $(seq 0 $LM);do M${each};done;} POS(){ if [[ $cur == up ]];then ((i--));fi     if [[ $cur == dn ]];then ((i++));fi     if [[ $i -lt 0 ]];then i=$LM;fi     if [[ $i -gt $LM ]];then i=0;fi;} REFRESH(){ after=$((i+1)); before=$((i-!))     if [[ $before -lt 0 ]];then before=$LM;fi     if [[ $after -gt 0 ]];then after=0;fi     if [[ $j -lt $i ]];then UNMARK;M$before;else UNMARK;M$after;fi     if [[ $after -eq 0 ]] || [$before -eq $LM ];then          UNMARK; M$before; M$after;fi;j=$i;UNMARK;M$before;M$after;} INIT(){ R;HEAD;FOOT;MENU;} SC(){ REFRESH;MARK;$S;$b;cur='ARROW';} ES(){ MARK;$e "ENTER = main menu ";$b;read;INIT;};INIT while [[ "$0" != " " ]]; do case $i in     0) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(w     )\n";ES;fi;;     1) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(ifconfig  )\n";ES;fi;;     2) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(df -h )\n";ES;fi;;     3) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(route -n  )\n";ES;fi;;     4) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS(date  )\n";ES;fi;;     5) S=M2;SC;if [[ $cur == "" ]];then R;$e "\nS($e by oTo )\n";ES;fi;;     6) S=M2;SC;if [[ $cur == "" ]];then R;exit 0;fi;;  esac;POS;done; convergence of $\sum_{n=1}^{\infty }\frac{(-1)^{[\sqrt{n}]}}{n}$ where $[a]$ means floor function of $a$ I have a problem with solving the convergence of $\sum_{n=1}^{\infty }\frac{(-1)^{[\sqrt{n}]}}{n}$ where $[a]$ means floor function of $a$. I've already made the first thep and rewrote it with the $\sum_{n=1}^{\infty }(-1)^{n}\sum_{k=n^2}^{(n+1)^2-1}\frac{1}{k}$, which converges if and only if the previous one converges. Now I want to use Leibniz criterion, but I cannot prove that the second sum in the previous expression is monotonic. Does anyone know how to do it? (I cannot use Riemann integral, we haven't defined it yet) What's the easiest way to install a missing Perl module? Error with \newcommand using \multicolumn in table I want to define a new command \tableheadline that (1) formats the headline text in some way and (2) takes an optional argument for spanning multiple columns while centering the headline.  I tried the following implementation:  \documentclass{minimal}  \newcommand{\tableheadline}[2][1]{\multicolumn{#1}{c}{\textit{#2}}} % \newcommand{\tableheadline}[2][1]{\multicolumn{1}{c}{\textit{#2}}}  \begin{document}   \begin{tabular}[\textwidth]{cc}      \tableheadline[2]{C}     \tableheadline{A} &amp; \tableheadline{B} \\     0 &amp; 1 \\     2 &amp; 3 \\   \end{tabular} \end{document}   This, however, gives me an error:  ! Misplaced \omit. \multispan -&gt;\omit                    \@multispan l.8 \tableheadline[2]{C}   I also get the error when I do not even use the optional argument (see commented line in the MWE). Any ideas what I am doing wrong? .NET Casting Generic List What is a good method for short text clustering? I am working on a text clustering problem. The data contains several sentences. Is there a good algorithm which reaches high accuracy on short text?  Can you provide good references?   Algorithms such as KMeans, spectral clustering does not work well for this problem. On exactly what criteria is the Electorate badge awarded? There is a badge named Electorate which is going to be awarded on "Voted on 600 questions and 25% or more of total votes are on questions". Can somebody explain this to me?  Voted on 600 Question and 25% are on Question? is this  means   Voted on 600 Posts and 25% are on Question  Can't SO place some easy text or Definitions for average users? How can I use a variable in the replacement side of the Perl substitution operator? I would like to do the following: $find = &quot;start (.*) end&quot;; $replace = &quot;foo \1 bar&quot;;  $var = &quot;start middle end&quot;; $var =~ s/$find/$replace/;  I would expect $var to contain &quot;foo middle bar&quot;, but it does not work. Neither does: $replace = 'foo \1 bar';  Somehow I am missing something regarding the escaping. Is there any way to view saved wifi passwords on the iPhone? Can a wizard under the effects of Feign Death issue telepathic commands to their familiar? A wizard has a summoned familiar within 100 feet that is currently idle and has  (PHB, p. 240) cast on them by another PC. Would the wizard still be able to issue telepathic commands to their familiar?  Specifically, while Feign Death states that they appear dead and are incapacitated and blind, they aren't listed as being unconscious.   doesn't state that issuing telepathic commands requires an action (which you can't do while incapacitated), whereas it does state that seeing through your familiar's eyes requires an action as does dismissing it.  I was DMing a game where this occurred and I hastily ruled that the wizard was unable to issue commands, but I am second guessing myself after the fact. And knowing my PCs this situation is likely to happen again. Inhumans vs. Mutants in the Marvel Universe What's the difference between those two? Aren't Inhumans supposed to be an evolution of the human race "forced" by the Kree? Which is almost the same as Mutants, the natural evolution of the human race...  What exactly am I missing? bash script – for i in {x..y} i am writing a script in bash and I would like to give the two numbers in the brackets, x and y to be precise, as variables.  I wrote the script this way:  echo "x?" read x echo "y" read y  for i in {$x..$y}   but it does not work.  How could I set it up to make it function?  Thanks a lot in advance. Suppose the mean noon-time temperature for September days in San Diego is 24∘ and the standard deviation is 4.9. (Temperature in this problem is measured in degrees celsius)  On September 26, 1963, the all-time record of noon-time temperature in San Diego of 44∘ was hit. Assume the temperature distribution is symmetric around the mean, what is the Chebyshev bound for the probability of breaking (or tieing) this record?  I am having a hard time understanding this. Could someone explain to me how to do this?