Show that :$89|2^{44}-1$ Using Fermat's theorem we have:    $2^{88}\equiv{1}\pmod{89}\ \Rightarrow\ (2^{44}-1)(2^{44}+1)=89k$ , now how can be sure that: $89|2^{44}-1$??
As the title, I was asked to show that $2^{44}-1$ is divisible by $89$.  My first attempt is to change the expression into $(2^{22}-1)(2^{22}+1)$.  Then further simplified it into $(2^{11}-1)(2^{11}+1)(2^{22}+1)$, I used my calculator and was able to show that $2^{11}-1$ is divisible by $89$ but then I don't know how to show it with modular arithmetic. I do think that it is quite similar to the form where we can use the Fermat's little theorem. $(\sqrt{2})^{88}-1$. (Though I do understand Flt can only be applied to integer.)  Can someone tell me whether I can take square root in modular arithmetic as well? I am still pretty new to the modular arithmetic. Thank you very much.
Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$? Iterating through list of list in Python Parse and Extract Attributes from JSON Limit connection rights on SQL Server I have an app to deploy in production that uses 'honor system' security. That is, all users connect to the DB using a SQL user/passwd credential and the app manages permissions itself. The latter part doesn't bother me as much as the fact that the connection object contains embedded credentials and can be copied around freely. I'm try to find some way to limit connections to a more limited set of clients. I can create firewall rules to limit by IP, of course. Is there any way to 'prequalify' SQL logins either by Machine account or domain membership? TeX capacity exceeded with glossary package My document contains many thousands of glossary items. When compiling with XeLaTeX, about 1/8 of the way through the document, it reports:  ! TeX capacity exceeded, sorry [main memory size=3000000]. \@glolist@ ...},{13586},{9406},{4539},{12811},{179                                                   51},{5831},{1047},{14948},    The numbers you see above are ones placed in #1 of \newglossaryentry{#1}{#2}. These unique numbers are assigned to each entry for convenience. I have successfully compiled short tests of the document, which only used 1% of the full content. The resulting .glo file is about 1 MB in size.   Could this be the result of some error on my part or am I giving TeX too much work to do?  Update:  I have followed the instructions below (edited texmf.cnf and running  fmtutil-sys --all. This has allowed me to compile an additional portion of the book, but no matter the settings I add, I can't go beyond 5/8 of the way through.  A compile with extra_mem_top.xelatex = 30000000 and extra_mem_top.xelatex = 50000000 both result in:  TeX capacity exceeded, sorry [pool size=3186676].   Update:  I have done the following:  $ kpsewhich texmf.cnf /usr/local/texlive/2011/texmf.cnf $ nano /usr/local/texlive/2011/texmf.cnf   To the texmf.cnf file, following instructions at , I added:  main_memory = 230000000 extra_mem_top = 10000000 extra_mem_bot = 10000000 save_size = 150000 stack_size = 150000  main_memory.xelatex = 30000000 extra_mem_top.xelatex = 30000000   After this, I tried: - fmtutil-sys --all, then compiling xelatex file.tex. - fmtutil --all, then compiling xelatex file.tex. - fmtutil-sys --byfmt xelatex, then compiling with xelatex file.tex. - texhash, then compiling with xelatex file.tex.  All of these resulted in no changes in results. I still get:  TeX capacity exceeded, sorry [pool size=3186676]. why is the output of the following code coming to be zero? Given a Java InputStream, how can I determine the current offset in the stream? Does a capacitor store charge? There're some easy concepts which I don't get exactly in my mind. I'm afraid I've been studying these things for two years of my engineering but they still bother me. Capacitor is one of them. Can someone explain?    What does a capacitor do? Does it store charges? If so, then how does it do?   I have searched it on Google and Yahoo but didn't find any helpful thing there (for me). So I will be glad if I got my problem solved here.  P.S. I hope that the question would not again be an off-topic, as it always does and moreover people don't suggest then where to go. It's a real sad thing. Browser back leaves page after going fullscreen in stack snippet Opening a Stack Snippet in full screen should use history.pushState() to add an entry to the browser history stack. Clicking back will then close the snippet rather than leave the page completely. Complex number isomorphic to certain $2\times 2$ matrices? I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is simple enough.  $$\phi:a+bi \rightarrow \begin{pmatrix} a &amp; -b \\ b &amp; a \end{pmatrix}$$  This is what I have so far:  Injective: I am also confused over the fact that there are two operations, and in turn two neutral elements (1 and 0). Showing that the kernel is trivial is usually the way I go about proving whether a mapping is injective, but I just can't grasp this.  $$ \phi(z_1) = \phi(z_2) \implies \phi(z_1)\phi(z_2)^{-1} = I = \phi(z_1)\phi(z_2^{-1}) = \phi(z_2)\phi(z_2^{-1}).$$  So if we can just show that the kernel of $\phi$ is trivial, then it also shows that $z_1 = z_2$. The only complex number that maps to the identity matrix is one where $a = 1$ and $b = 0$, $a + bi = 1 + 0i = 1$.  Using a similar argument for addition we can just say that the only complex number $z$ such that $\phi(z) = 0\text{-matrix}$, is one where $a=0$ and $b=0$, $a+bi=0+0i=0$.   Surjective:  I forgot to add this before I posted, but I honestly don't really understand how to prove this because it just seems so obvious. All possible $2\times2$ matrices of that form have a complex representation because the complex number can always be identified by its real parts and since the elements of the $2\times2$ matrix are real then the mapping is obviously onto.  I have always had trouble understanding when I can say that I have "rigorously" proved something, so any help would be appreciated! Where can we find frequently updated details about travel/entry restrictions due to Coronavirus (nCoV-19)? Django Template Variables and Javascript When I render a page using the Django template renderer, I can pass in a dictionary variable containing various values to manipulate them in the page using {{ myVar }}.  Is there a way to access the same variable in Javascript (perhaps using the DOM, I don't know how Django makes the variables accessible)? I want to be able to lookup details using an AJAX lookup based on the values contained in the variables passed in. Eigenvalues and power of a matrix DEFINE vs Variable in PHP I am trying to follow the best practice design patterns for triggers but get the error Constructor not defined: [CaseTriggerHandler].(). What am I missing?  Trigger  trigger CaseTrigger on Case (before insert, before update, before delete, after insert, after update, after delete, after undelete) {      CaseTriggerHandler handler = new CaseTriggerHandler();      //Before Insert     if(Trigger.isInsert &amp;&amp; Trigger.isBefore){         handler.OnBeforeInsert(Trigger.new);     }     //After Insert     else if(Trigger.isInsert &amp;&amp; Trigger.isAfter){         handler.OnAfterInsert(Trigger.new);     }     //Before Update     else if(Trigger.isUpdate &amp;&amp; Trigger.isBefore){         handler.OnBeforeUpdate(Trigger.old, Trigger.new, Trigger.newMap);     }     //After Update     else if(Trigger.isUpdate &amp;&amp; Trigger.isAfter){         handler.OnAfterUpdate(Trigger.old, Trigger.new, Trigger.newMap);     }     //Before Delete     else if(Trigger.isDelete &amp;&amp; Trigger.isBefore){         handler.OnBeforeDelete(Trigger.old, Trigger.oldMap);     }     //After Delete     else if(Trigger.isDelete &amp;&amp; Trigger.isAfter){         handler.OnAfterDelete(Trigger.old, Trigger.oldMap);     }     //After Undelete     else if(Trigger.isUnDelete){         handler.OnUndelete(Trigger.new);     } }   Handler      public with sharing class CaseTriggerHandler {      private boolean isExecuting = false;      public CaseTriggerHandler(boolean isExecuting){         this.isExecuting = isExecuting;     }      public void OnBeforeInsert(List&lt;Case&gt; newCases){         //EXECUTE BEFORE INSERT LOGIC     }      public void OnAfterInsert(List&lt;Case&gt; newCases){         //EXECUTE AFTER INSERT LOGIC     }      public void OnBeforeUpdate(List&lt;Case&gt; oldCases, List&lt;Case&gt; updatedCases, Map&lt;Id, Case&gt; CaseMap){         //BEFORE UPDATE LOGIC     }      public void OnAfterUpdate(List&lt;Case&gt; oldCases, List&lt;Case&gt; updatedCases, Map&lt;Id, Case&gt; CaseMap){         //AFTER UPDATE LOGIC     }      public void OnBeforeDelete(List&lt;Case&gt; CasesToDelete, Map&lt;Id, Case&gt; CaseMap){         //BEFORE DELETE LOGIC     }      public void OnAfterDelete(List&lt;Case&gt; deletedCases, Map&lt;Id, Case&gt; CaseMap){         //AFTER DELETE LOGIC     }      public void OnUndelete(List&lt;Case&gt; restoredCases){         //AFTER UNDELETE LOGIC     }      public boolean IsTriggerContext{         get{ return isExecuting;}     } } Biblatex enumerating sorted bibliography (using authoryear-ibid style) How can I set biblatex to insert "1. " before first bibliography item and so on? The bibliography is sorted by author and the numbers are not meant to be used anywhere else. External monitors show blank screen until restart MacBook Pro (15-inch, 2016) with 2 USB-C to HDMI adapters. The setup works great if I start up in that configuration.   If I connect the adapters while the machine is running or simply sleeping, one of the monitors will never get a signal and stay in low power mode. The OS sees the monitor, it shows in the display arrangement settings and I can move the mouse into it, but until I restart, I can't see anything on it.  Once I have restarted I can let the machine sleep or whatever, and the display still works unless I disconnect the adapter. As soon as I do that a restart is required to use it again.   Using MacOS Sierra (10.12.4)  Using this adapter: The direction of static friction? Is it possible to set start-up applications to initialize in certain workspaces? I'd like to not have to manually distribute my start-up applications across my numerous workspaces every time I boot up my machine. I'm running Ubuntu 12.04 .  Any Help ? After installing Adobe PDF reader, it opens files with the .PDF file extension. Ones' default web browser opens .html files and notepad opens .txt files, etc.  Where exactly in the registry are these filetype - program associations stored in the registry?   In other words, how does Windows know with which software/program it must open a certain file(type)?