$n$ is a natural number and $p$ is prime.  I have seen the question that is marked as duplicate, but it does not awnser my question.  This is what I tried: First suppose $n\equiv 1$ mod $q$. Then $1 + n+\dots n^{q-1}\equiv 0$ mod $ q$. So there's an $a$ with $ap\equiv 0$ mod $q$. From here I want to show $p=q$, but don't now how to continue.  Now suppose $n\not\equiv 1$ mod $q$. We know $p(n-1)\mid (n^q - 1)$, so there's an $a$ with $ap(n-1) = (n^q -1)$. Using Fermat's little Theorem we see $ap\equiv 1$ mod $q$. From here I want to show $p\equiv 1$ mod $q$, but again don't know how to continue.  I would appreciate any help.
let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$. Prove that $p = 1 \mod q$ or $p = q$.  I don't actually know how to start, i thought of using fermat's little theorem or an application of eulers formula, but it didn't really give me any insights.   Any hints?
SQL Server FOR XML Enclosing Element? Drush hangs when trying to bootstrap Drupal Security of GCM if Implemented with Stream Cipher instead of Block Cipher Is there a travelling speed of for electric field? If yes, what is it? Consider two circuits containing a battery, bulb, switch and conducting wires but of length 1 meter in one case and 1000KM in other. When switched on at the same time both the bulbs glow instantaneously. I think it is because, electric filed travels from positive terminal of battery to negative terminal through wire almost instantaneously.  My question is there any speed for this electric field? If yes, what is it? By any modification in this circuit is it possible to induce any delay in glowing the bulb? I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.  I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a&amp; b\\ c&amp; d \end{array}\right)$$ and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution to $A = SBS^{-1}$. That worked fine, but what will I do if I have $3\times 3$ or $9\times 9$ matrices? I can't possibly make system that complex and solve it.  How can I know if any two matrices represent the "same" linear transformation with different bases?  That is, how can I find $S$ that change of basis matrix?  I tried making $A$ and $B$ into linear transformations... but without the bases for the linear transformations I had no way of comparing them.  (I have read that similar matrices will have the same eigenvalues... and the same "trace" --but my class has not studied these yet. Also, it may be the case that some matrices with the same trace and eigenvalues are not similar so this will not solve my problem.)  I have one idea. Maybe if I look at the reduced col. and row echelon forms that will tell me something about the basis for the linear transformation? I'm not really certain how this would work though? Please help. What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.  The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like that.  But when it comes to vectors $\vec{a} \cdot \vec{b}$, I'm not sure what to say. "It is $\vec{a}$ added to itself $\vec{b}$ times" which doesn't make much sense to me. Is it fair to buy a return ticket I know I will never use just because it's cheaper? I am planning a trip to Asia and then back to Rio. The thing is that I am traveling on flexible dates, so I am thinking of buying a one way ticket to Hong Kong, and I was planning to do the same when I want to come back.  But what I noticed is that, for example, a one way ticket from Melbourne to Rio costs no less than $2700 dollars for the month of May.  When I do a search for a round-trip from Melbourne to Rio and back, I can find flights for $2200.  I know I will not be using that return to Melbourne, but I'd still prefer to save those $500.   Can there be any issue with buying a round-trip with a return that I will never use? Calculating a spherical polygon centroid Two column float with [t] using the cas-dc template I am using the cas-dc template for Elsevier. When trying to use \begin{figure*}[t], except I get the error 'The module 'cas/fig' does not have a key called 'cas/fig/t'. Other templates I use work fine. I have included \usepackage{stfloats}. I don't see anything mentioned that strictly forbids it, unless I missed this. Anyone know why? I use a 7805 for a project where the circuit needs a higher current (~2.8&nbsp;A) at 5&nbsp;V. So I assume that if I use both ICs in parallel I can increase the maximum current capacity. Would it work? How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit? Fred and George stumbled upon the Marauder's map in their first year. They passed it on to Harry while in their fifth year. Also note that, while talking to Harry on the train, Ron described Scabbers as being "Percy's old rat".   Hence, we can safely assume that the twins had the Marauder's Map for about five years. How could they not see Peter Pettigrew on the map, with Percy for first two years and with Ron for the next three years?  How come Harry didn't see Peter? How was Lupin the only one able to see Peter on the map? Non-EU spouse of UK citizen Schengen visa requirements I'm a British citizen and my wife is in the UK on a spouse visa. She holds a BRP card. We're planning to visit Germany and some other Schengen countries, and I've got a question around visa requirements/supporting documents.  I've read in few places (i.e ) that the visa requirements are more relaxed if the applicant is a spouse of a UK citizen, and is travelling with the UK citizen.  How relaxed are the requirements? Does my wife need to submit supporting documents for accommodation, finances, flight booking, etc.? Or is it simply submitting the application form alongside the passports and BRP? (i.e standard documentation for Schengen visa application section from ) Can Quantum Entanglement and Quantum Superposition be considered the same phenomenon? Quantum entanglement is known to be the exchange of quantum information between two particles at a distance, while quantum superposition is known to be the uncertainty of a particle (or particles) being in several states at once (which could also involve the exchange of quantum information for a particle that is known to be in several locations simultaneously).  I was wondering if all of this was nothing more but the exchange of quantum information between different masses, and if this could clear up all the confusion in terms of how quantum systems connect in this field of science.  A clear explanation for how both of these quantum phenomena work, and if they really are connected (the exchange of quantum information?) would be much appreciated. A generic list of anonymous class In C# 3.0 you can create anonymous class with the following syntax  var o = new { Id = 1, Name = "Foo" };   Is there a way to add these anonymous class to a generic list?  Example:  var o = new { Id = 1, Name = "Foo" }; var o1 = new { Id = 2, Name = "Bar" };  List&lt;var&gt; list = new List&lt;var&gt;(); list.Add(o); list.Add(o1);   Another Example:  List&lt;var&gt; list = new List&lt;var&gt;();  while (....) {     ....     list.Add(new {Id = x, Name = y});     .... } What can I do if a repository/PPA does not have a Release file? Newton's cradle What's the difference between JavaScript and Java? What's the difference between JavaScript and Java? I have a command that I want to have run again automatically each time it terminates, so I ran something like this:  while [ 1 ]; do COMMAND; done;   but if I can't stop the loop with Ctrl-c as that just kills COMMAND and not the entire loop.  How would I achieve something similar but which I can stop without having to close the terminal? $X$ is homeomorphic to $X\times X$ (TIFR GS $2014$) Question is :  Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:   $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above.   I guess first two options are false.  We do have possibility that product of two connected spaces is connected.  So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.  We do have possibility that product of two compact spaces is compact.  So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.  I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.  So, only thing I have problem with is third option.  I could do nothing for that third option..  I would be thankful if some one can help me out to clear this.  Thank you :)