How to derive the chromatic polynomial from a Cycle? I derived the chromatic polynomial for a triangle $ K_3$ it's: $t(t-1)(t-2)$ But I don't understand how to get it for Cycles $C_n$.
This is a homework question. But I am completely stuck.  My only intuition was to go about it inductively from a "greedy algorithm" maybe know as the deletion-contraction algorithm. And to somehow use the information about the jth cycle to solve the j+1th. But I'm not sure how I'd do it. Thank you very much for looking this over.
We can prove that there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1$. This leads to the following question:     Is it true that for every pair of primes $p&lt;q$ there are only finitely many integers $n$ such that $p^n-1\mid q^n-1$? Are there two primes $p&lt;q$ and an integer $n&gt;p+q$ such that $p^n-1\mid q^n-1$?Is it true that if $n&gt;6$ then $2^n-1\nmid 5^n-1$?   Edit: Here are some examples: \begin{array}{ll} 2^{36}-1\mid 41^{36}-1, &amp;3^{12}-1\mid 97^{12}-1,\\ 5^{6}-1\mid 37^{6}-1, &amp;7^{4}-1\mid 151^{4}-1. \end{array}  Now I prove there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1.$  Proof: Denote $A=2^n-1$ and $B=3^n-1$. If $n$ is even then $3$ divides $A$ but not $B$, a contradiction.  If $n$ is odd then $A\equiv -5\pmod {12}$. Since every prime greater than $3$ is $\equiv \pm1,\pm5 \pmod {12}$, some prime factor $p$ of $A$ must be congruent to $\pm5 \pmod {12}$. As $p \mid B$, we have $3^{n+1}\equiv 3 \pmod p$. Since $n+1$ is even, we get $(\frac{3}{p})=1$, a contradiction again. iPhone 6s with iOS 12.4 not connecting to MacBook Pro mid2015 with macOS 10.14.6 I am using an iPhone 6s with iOS 12.4 and a MacBook Pro mid2015 with macOS 10.14.6. Since recently (I am not exactly sure when it started), I cannot charge the phone with my computer anymore. As soon as I plug the phone in, it will play the charging sound repeatedly. The right-hand side port sometimes works, the lefthand side port never works. When I connect my old iPhone 5 (using the same cable), it charges without any issues. My iPad mini 4 shows the same issue.  Sometimes the "Do you trust this computer" setting flashes on the phone and disappears before I can click anything.   How can I fix this?  Update 1  Sometimes I get the Software Update dialogue in macOS. After approx. 2 minutes it shows the following:     Update 2  For some reason the issue disappeared for a while but is back now. Disabling "Settings" > "Touch ID &amp; Passcode" > "USB Accessoires" seems like it makes the problem a bit less bad, but not go away.  Update 3  This is driving me crazy. I upgraded to Mojave 10.14.3 mostly to get rid of the issue. It persists. Some more random information: Before logging in (Filevault pre-boot authentication), the phone will charge fine. Only when a user account is loaded, the issue appears. The likelihood of a successful connection increases with the battery state of the iPhone. Workaround 2 in my answer below suggest that this a "Trust issue" from the iPhone's side. I suppose the solution will have to be hidden somewhere there...    What I tried so far (will update):   Reset Privacy settings on iPhone Different (original, almost new Apple) lightning cables   Remove and add "iPhone USB" in Network Settings  SMC reset (), PRAM reset Upgrade to latest iOS 12.4 and MacOS 10.14.6 An unclear sentence in the book "Griffiths, Harris-Principles of algebraic geometry" In Dark Souls 2 there is a 'Shrine of Winter' near Sinners Rise. When you approach the shrine it will not let you open the door and a message pops up that says      Seek Mightier Souls   What does this mean and how do you open this shrine? Please revoke badges gained by abusive voting In  (not feature request), :  ... if we find that people are abusing the system their accounts may get suspended. I would consider that kind of behavior abuse.  Well  sure qualifies. He has 11 Stack Exchange accounts and has in the last day spammed at least 8 of them to gain the Suffrage and sometimes Vox Populi badges. That's over 250 votes† in just a few hours! (And  -- while voting on other user's posts too.) I guarantee that he's not voting on those posts for/by their merits. Here's when he scammed the Suffrage badge per site (while also gaming the Vox Populi on some sites):                      - 2017-03-11 16:23:15Z                     - 2017-03-11 17:41:41Z                     - 2017-03-11 18:08:13Z                           - 2017-03-11 19:25:10Z                  - 2017-03-11 19:55:16Z                               - 2017-03-11 20:00:49Z               - 2017-03-11 20:03:13Z  - 2017-03-11 21:11:40Z  Note that he got the badge on three sites within an 8 minute span (but that might just be job run times). Please revoke badges gained through improper voting. And, maybe also warn/suspend such abusive voters.   † His account vote totals don't show all votes as some serial voting was reversed -- but none of the ill-gotten badges were. Search for singleton tags on questions What is $i^i$ and $(i^i)^i$ What is $i^i$ and $(i^i)^i$. I fond some answers  but is there a clearer explanation of this? Maybe something help me to understand. how to remove object from NSDictionary Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? I am planning to travel to India (COK airport, Kerala) from United States. I want to take my laptop as well as my tablet (iPad). Am I allowed to take both in my backpack with my documents related my work. I think I am allowed to take only one computer as per the rule. Will the customs consider my iPad as a computer?  What all I can take?   One of my friends took 2 laptops (1 work and 1 personal) and had trouble at the airport.  Also I am planning to take some wristwatches (7 of them, each avg price is 45 $). I read in a website that I can take stuff up to max 35000 INR. Is that true? In that case, how will I manage my stuff? When are infinities okay? I'm a little confused about how to handle infinities in physics. On the one hand, I always here things like infinities are not physically tenable answers. GR can't answer anything about the origin of the universe because of singularities. Throw away the radial solution that gives an infinite potential when solving the Laplace equation in spherical coordinates. But then there are other times that the infinities are okay. For instance, it is usually okay to set the reference point for your potential at infinity. There are certain times in David Tong's quantum field theory lecture notes where he seems to almost to hand wave away infinities. So, philosophically, physically, and mathematically when are infinities okay, and when are they not okay? Frequency of the sound when blowing in a bottle Install CentOS 7 on HP dl120 gen9 server with b140i raid controller I'm trying to install CentOS 7 on a HP DL120 gen 9 server with a b140i controller. I have 4 1TB HDD setup in a raid 1+0 and the CentOS installer doesn't seem to recognize the logical drive created from the storage utility.  From my understanding i need some drivers for CentOS 7 to recognize the B140i controller. Also, the driver seems to be closed source by HP.  What is my approach here? Is there any other free linux OS that i can use on this server and that would recognize the raid controller?  Thank you. macOS: Is there a way to show an image constantly on screen (like a watermark)? I'm looking for a very simple way to add a watermark to my screen which should be shown whatever I'm doing, is it during a Zoom call, or is it when giving a presentation, or similar (I don't necessarily need it to show during fullscreen mode). The reason is that I want to record my screen using QuickTime Player, and the watermark should be on the video. I know that there are many post-production tools that offer adding watermarks, but I don't want to do any post-production, I want to have everything on screen while recording already. (I also know that there are screen recorders which offer placing a watermark to the end result. I'd rather also avoid using another screen recording software, but if I don't find any other way to do it, then I'm open to a simple, non-pricy alternative to QuickTime Player.) I have a webpage that implements a set of tabs each showing different content. The tab clicks do not refresh the page but hide/unhide contents at the client side.  Now there is a requirement to change the page title according to the tab selected on the page ( for SEO reasons ). Is this possible? Can someone suggest a solution to dynamically alter the page title via javascript without reloading the page? Problem:  Ajax suggest-search on [n] ingredients in recipes. That is: match recipes against multiple ingredients.  For instance: SELECT Recipes using "flower", "salt" would produce: "Pizza", "Bread", "Saltwater" and so forth.  Tables:  Ingredients [     IngredientsID INT [PK],     IngredientsName VARCHAR ]  Recipes [     RecipesID INT [PK],     RecipesName VARCHAR ]  IngredientsRecipes [     IngredientsRecipesID INT [PK],     IngredientsID INT,     RecipesID INT ]   Query:  SELECT     Recipes.RecipesID,     Recipes.RecipesName,     Ingredients.IngredientsID,     Ingredients.IngredientsName FROM     IngredientsRecipes      INNER JOIN Ingredients     ON IngredientsRecipes.IngredientsID = Ingredients.IngredientsID      INNER JOIN Recipes     ON IngredientsRecipes.RecipesID = Recipes.RecipesID WHERE     Ingredients.IngredientsName IN ('salt', 'water', 'flower')   I am currently constructing my query using ASP.NET C# because of the dynamic nature of the WHERE clause.  I bites that I have to construct the query in my code-layer instead of using a stored procedure/pure SQL, which in theory should be much faster.  Have you guys got any thoughts on how I would move all of the logic from my code-layer to pure SQL, or at least how I can optimize the performance of what I'm doing?  I am thinking along the lines of temporary tables:  Step one: SELECT IngredientsID FROM Ingredients and INSERT INTO temp-table  Step two: SELECT RecipesName FROM Recipes joined with IngredientsRecipes joined with temp-table.IngredientsID what is the area of the region [ (x,y):0<x , y<1 , 3/4<x+y<3/2 ]? An idiom for deriving pleasure from another's suffering I believe it is what the Germans call "". English itself has no such equivalent word. (Although it has been adopted as a .)  Does an idiom exist that describes it? 'innerText' works in IE, but not in Firefox Help with math symbols in BibTeX I am writing a PhD thesis using LaTeX. Every thing is going fine, but I have problem with the output of the mathematical symbols in the references lists. I copy one example from the Pdf file to make it clear for you. You can see that the fraction remains the same.      [1] Mezzacapo, F. and Cirac, J. I. (2010) Ground-state properties of   the spin-\frac{1}{2} antiferromagnetic Heisenberg model on the   triangular lattice: a variational study based on entangled-plaquette   states. New. J. Phys., 12,   103039.   You can see below the all the packages I use:  \documentclass[12pt,PhD,twoside,a4paper]{GB_muthesis}%[10pt,PhD, singlespace] \usepackage[pdftex]{graphicx} \usepackage[bookmarks=true,colorlinks=true,linkcolor=blue, urlcolor=blue,citecolor=blue]{hyperref} \usepackage[numbers,comma, sort&amp;compress]{natbib} \usepackage{amsmath} \usepackage{amssymb} \usepackage{mathtools} \usepackage{rotating} %\usepackage{GE_manthesis} \usepackage{GB_thesis} \usepackage{bm} \usepackage{fmtcount} \usepackage{setspace} \usepackage{chapterbib} \usepackage{subfigure}% allow subfigures \usepackage{amsfonts} \usepackage{url, enumitem} \usepackage{lscape} \usepackage{notoccite}%This package is for ingnoring the citation in the caption of the figures and tables. \begin{document}