Let p and q are odd prime numbers such that q=2p+1. Prove that   i) $-4$ is primitive root modulo $q$.  ii) If $p  \equiv  1 \pmod 4$ then $2$ is a primitive root mod $q$.   iii) If $p  \equiv  -1 \pmod 4$ then $-2$ is a primitive root mod $q$.
If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.   I cannot get heads nor tails of how to even start this let alone finish it
Forgive me for the unclear title. Some time ago, my HTC M8, which is rooted and runs on a  showed something weird. Firstly, in the Google Maps App a mysterious blue dot appeared, that captures the screen completely and makes it impossible to quit or use the application. It looks like this (I could make a screenshot)     When this happens in Maps, the only thing I can do is a hard restart with Power+Volume Up.  Since I have seen this occasionally in the camera app as well, I suspect it is not a bug of Maps itself. I know that the Venom ROM is not standard but maybe this is connected to some setting and anyone has seen this. Btw, the Venom ran smoothly for almost 3 years now.  Question: Has someone a clue what this could be and how I can get rid of it? What it's the best way to generate a string of \t's in C#  I am learning C# and experimenting with different ways of saying the same thing.    Tabs(uint t) is a function that returns a string with t amount of \t's  For example Tabs(3) returns "\t\t\t"  Which of these three ways of implementing Tabs(uint numTabs) is best?  Of course that depends on what "best" means.     The LINQ version is only two lines, which is nice.  But are the calls to Repeat and Aggregate unnecessarily time/resource consuming? The StringBuilder version is very clear but is the StringBuilder class somehow slower? The string version is basic, which means it is easy to understand.   Does it not matter at all?  Are they all equal?   These are all questions to help me get a better feel for C#.     private string Tabs(uint numTabs) {     IEnumerable&lt;string&gt; tabs = Enumerable.Repeat("\t", (int) numTabs);     return (numTabs &gt; 0) ? tabs.Aggregate((sum, next) =&gt; sum + next) : "";  }    private string Tabs(uint numTabs) {     StringBuilder sb = new StringBuilder();     for (uint i = 0; i &lt; numTabs; i++)         sb.Append("\t");      return sb.ToString(); }    private string Tabs(uint numTabs) {     string output = "";     for (uint i = 0; i &lt; numTabs; i++)     {         output += '\t';     }     return output;  } What is the difference between metonymy and synecdoche? There seems to be a preference for to have the    18 mm - 55 mm 55 mm - 250 mm   lenses, but no mention of why they're better than a single 18-200mm. Could someone please explain why the two separate are better than the one combined, and if anyone has quantified this difference?  (I'm curious about Canon in particular.) Reviewed suggested edit view for title-only edits takes body of last revision? Yesterday I was pointed at some rejected suggested edits. Some of the corrections were valid, so I edited the posts accordingly, while making several further edits. Looking at the suggested edits now however, it seems that for those posts where only changes to the title were originally suggested, the post mortem suggested edit view now incorporates the body of the latest revision. (Or at least it appears like that to me)  Take for example . If you take a look at  of that post, you will see my later edits. The edited body seems to have been incorporated in the suggested edit view. For edits where originally changes to the body were suggested, there is no such confusion.   Why does this matter? In the case of some post mortem discussions or explanations it might help to see the actual situation as it was when the edit was suggested. Especially in the case of edits that might be considered too minor.   So have I stumbled upon a bug here? Can I recover app data after the phone has been Factory Reset? How to make Gnome's "Activities" bar always visible? I just installed Gnome, and when I click on "Activities" a convenient bar shows up from the left, allowing me to switch to other applications. Unfortunately, the bar hides as soon as I choose an application.  How to make this bar always visible? If in Pete's World there's no Doctor, how is there a Torchwood? How can I ensure my trip to the UK will not have to be cancelled because of Brexit? I've got a trip planned to the UK in June. Due to brexit, I'm worried what changes may be introduced before my arrival.  I'm traveling on a EU / Polish passport. I'm going for a couple weeks of holiday.  Is there anything I can do right now to make it less likely any changes won't stop me at the immigration? For example, applying for some tourist visa which does not rely on the EU status? How do I find the caller of a method using stacktrace or reflection? I need to find the caller of a method. Is it possible using stacktrace or reflection? I was browsing PhD position offers in Europe on different websites and I was quite surprised to find that several offers contained sentences such as     University of X value equality and diversity. We strongly encourage women and BAME applicants for this position.    where BAME stands for Black, Asian and Minority Ethnicities; or     The research group aims to increase the number of women in scientific positions. Female candidates are therefore encouraged to apply.   Is it considered normal in the academic environment to discriminate between  candidates according to their race and/or gender for "egalitarian" reasons?   How is the choice of a candidate based on their gender and/or race considered "egalitarian"? What is the difference between weight and mass? Probability of correctly guessing 12 zodiac signs without replacement I just watched  where someone assigns a zodiac sign to twelve different individuals. There is one of each sign.  My question, Guessing at random, (this was not quite at random), what number of people are you expected to guess correctly? And with what probability are you expected to guess a given number of people correctly?  i.e. What are the odds of getting 1, 2, 3, 4, etc. correct. a 2d planet in the center, where the players will be able to walk on it and also will be affected by center gravity. planet will be static and there won't be any other planets. so;  Is there a way that I can change the gravity position? yes you can adjust gravity via project settings but I want it to be in the middle x,y,z(0,0,0)  I have tried creating my own gravity but it wasn't very useful because objects wouldnt fall, stand on foot and also in contact with another object they would vibrate instead of standing still. (if they stand on the planet they would vibrate)          private void FixedUpdate() {     foreach(GameObject item in grav)     {         Attact(item);     } }  private void Attact(GameObject gmf) {     Rigidbody2D rgg = gmf.GetComponent&lt;Rigidbody2D&gt;();      Vector3 direction = rg2d.position - rgg.position;     float distance = direction.sqrMagnitude;      float forceMag = (10 * rgg.mass) / distance;     Vector3 force = direction.normalized * forceMag * 10;      rgg.AddForce(force); }   I can create a script that will change the gravity by the opposite of the object's position. this does work but I cannot see any way to do it for more than one object. I wrote a Table Valued Function in Microsoft SQL Server 2008 to take a comma delimited column in a database to spit out separate rows for each value.  Ex: "one,two,three,four" would return a new table with only one column containing the following values:  one two three four   Does this code look error prone to you guys? When I test it with   SELECT * FROM utvf_Split('one,two,three,four',',')    it just runs forever and never returns anything. This is getting really disheartening especially since there are no built in split functions on MSSQL server (WHY WHY WHY?!) and all the similar functions I've found on the web are absolute trash or simply irrelevant to what I'm trying to do.  Here is the function:  USE *myDBname* GO SET ANSI_NULLS ON GO SET QUOTED_IDENTIFIER ON GO ALTER FUNCTION [dbo].[utvf_SPlit] (@String VARCHAR(MAX), @delimiter CHAR)  RETURNS @SplitValues TABLE (     Asset_ID VARCHAR(MAX) NOT NULL )  AS BEGIN             DECLARE @FoundIndex INT             DECLARE @ReturnValue VARCHAR(MAX)              SET @FoundIndex = CHARINDEX(@delimiter, @String)              WHILE (@FoundIndex &lt;&gt; 0)             BEGIN                   DECLARE @NextFoundIndex INT                   SET @NextFoundIndex = CHARINDEX(@delimiter, @String, @FoundIndex+1)                   SET @ReturnValue = SUBSTRING(@String, @FoundIndex,@NextFoundIndex-@FoundIndex)                   SET @FoundIndex = CHARINDEX(@delimiter, @String)                   INSERT @SplitValues (Asset_ID) VALUES (@ReturnValue)             END              RETURN END How should I parse user input in a text adventure game? Parsing user commands in a text adventure is a spectrum from 's simple "go north" to some mind-bogglingly clever ones in .  I seem to remember reading nice how-tos in computer magazines back in the 80s, but now I find almost nothing on the 'net except a .  How would you do it?    Update: I went with the simplest approach possible in . Compile nested Sums Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then it is irreducible. Are there any others? Consider the real-valued function  $f(x)=x^r$ where $r$ is a real number.  (1) For what values of $x$ and $r$, is this function differentiable?  (2) How do you prove the power rule, when $f(x)$ is differentiable?  It would be nice if the proof is accessible to a 12th grade student.  EDIT: Perhaps, I have not been very clear about my question.  I am not satisfied with the "textbook proofs" I have seen.  Please established the existence (of the derivative) first before proving the power rule.  That is, please consider first question first.  This is the reason why I call the question a mathematical engineering question.  (If we can we bring high level math down to Grade 12 level, without loosing precision, then I call it good mathematical engineering.) Are human bodies programmed to die? Following from this question: :  Is there any evidence that human bodies have systemic self-destruction built into their developmental program? I'm not talking about the cell death response, which I know is an important part of growth, development and cancer prevention.  I've read some things about telomere-shortening but don't know if this is a cause or an effect.