Let $p$ be a prime. I want to show that $p | a^p - a$. I want to use induction.   I showed that this is true for the case $a = 1$. I'm having trouble with the bridge.
Its very easy to prove $p\mid n^p-n$ for p=3,5,7, it fails for p=9 because $$     (n+1)^9-(n+1)= n^9+9n^8+36n^7+84n^6+126n^5+126n^4+84      n^3+36n^2+8n $$ and $84= 2²\times 3\times7$. Is it true for any $p$ prime? I have checked many primes, but have no idea for the general case.
Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$ Reference request: convergence for periodic Markov chains Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains? What does accepting an answer actually convey (2015 edition)? Parsing date/time string with timezone abbreviated name in Python? How to print the full NumPy array, without truncation? Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$   I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$  We have $13\mid n^{13}-n$, by Fermat's Little Theorem.  We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is odd.  And the numbers $5$ and $7$, how to proceed? Plotting graphs using bash Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant. How do you show that if $f$ is entire and if $\text{Im} f\ge0$, then $\text{Im} f$ is constant? Is this even true in the first place? Code coverage for wrapper class is 0% There should be infinitely many primes of the form $5+6n$. How do you prove it? The same should be true for $7+6n$. How to force the browser to reload cached CSS and JavaScript files I have noticed that some browsers (in particular, Firefox and ) are very zealous in using cached copies of .css and .js files, even between browser sessions. This leads to a problem when you update one of these files, but the user's browser keeps on using the cached copy. What is the most elegant way of forcing the user's browser to reload the file when it has changed? Ideally, the solution would not force the browser to reload the file on every visit to the page.  I have found  and  suggestion to be useful. It turns out there is a term for this: auto-versioning. I have posted a new answer below which is a combination of my original solution and John's suggestion. Another idea that was suggested by SCdF would be to append a bogus query string to the file. (Some Python code, to automatically use the timestamp as a bogus query string, was .) However, there is some discussion as to whether or not the browser would cache a file with a query string. (Remember, we want the browser to cache the file and use it on future visits. We only want it to fetch the file again when it has changed.) Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$ I got this question in my maths paper     Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$   and find the sum if it exists.   I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you. About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$ Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this? Build a basic Python iterator Is it unreasonable to assign a MySQL database to each user on my site? I'm creating a user-based website. For each user, I'll need a few MySQL tables to store different types of information (that is, userInfo, quotesSubmitted, and ratesSubmitted).  Is it a better idea to:  a) Create one database for the site (that is, "mySite") and then hundreds or thousands of tables inside this (that is, "userInfo_bob", "quotessubmitted_bob", "userInfo_shelly", and"quotesSubmitted_shelly")  or  b) Create hundreds or thousands of databases (that is, "Bob", "Shelly", etc.) and only a couple tables per database (that is, Inside of "Bob": userInfo, quotesSubmitted, ratesSubmitted, etc.)  Should I use one database, and many tables in that database, or many databases and few tables per database?    Edit:  The problem is that I need to keep track of who has rated what.  That means if a user has rated 300 quotes, I need to be able to know exactly which quotes the user has rated.  Maybe I should do this?  One table for quotes. One table to list users. One table to document ALL ratings that have been made (that is, Three columns: User, Quote, rating).  That seems reasonable. Is there any problem with that? I tried to add a new repository to software sources, but "add source" button is unavailable. I can't click on it.  I also tried to add a new software source from the terminal, but it didn't work.  sudo‬‬ ‫‪add-apt-repository‬‬ ‫‪ppa:n-muench/burg‬‬   s‫‪udo‬‬: command not found     I also tried these commands:  ‫‪sudo‬‬ ‫‪add-apt-repository‬‬ ‫‪deb‬‬ ‫‪http://ppa.launchpad.net/n-muench/burg/ubuntu‬‬ natty‬‬ ‫‪main‬‬   sudo‬‬: command not found Graph of a continuous function is closed How To check Windows device is rooted/jailbrake or not programatically? I was wondering if it was possible to use any of the API's provided by SF to build an external (C#, Java) application to manage building changesets.   Building changesets is very tedious when trying to include 100's of items and I was hoping to streamline the process for our team by building an application I can load an excel file to with the required components and have it build the outbound changeset.