Solving an equation with least residues and roots modulo $n$ I am facing the problem of proving $z^2 \equiv 1\pmod 8$ for all odd $z\in\mathbb{N}$. What I know is that for each $z \in \mathbb{N}$, $z = m^2-1$, for $m\in\mathbb{N}$. What I also know is that in a least residue system there are the following solutions for all $z\in\mathbb{N}$. The solutions are: $$n \equiv 1,3,5,7$$ Which lead to solutions of the initial problem: $$n = 8 \cdot k + 1\\ n = 8 \cdot k + 3 \\ n = 8 \cdot k + 5 \\ n = 8 \cdot k + 7$$ where $k\in\mathbb{N}$. My question now is how I can transfer my knowledge about the solutions for all $n\in\mathbb{N}$ to the key problem where $n$ must be an odd number and the proofs needs to be done for all odd $n$. I also know that $8 = 2^3$ which leads to $$(2\;\cdot\;m\;-\;1)^2\;\equiv\;1\;\pmod 8$$ $$(2\;\cdot\;m\;-\;1)^2\;\equiv\;1\;\pmod {2^3}$$ but I do not know what this tells me right now.
For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
So I understand that this has happened in the past: , but all of the &quot;duplicate&quot; posts are marked as  so here is a new bug report: On the , I am listed twice:  While that's fun and all, it isn't exactly right. It's a shame to type out an entire question, press post and then:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Could you put some sort of notice on the page itself before submitting it to let you know the question won't be going through?  Or perhaps tell me how many questions I've asked in the last 30 days if it's >0?  Edit  This is not a complaint or statement about the limit, just a request that the error handling be done in a slightly better way to save people a lot of time. Should we avoid negative words when writing error messages? Since the Earth has a magnetic field, can it, in theory, be run through a conductive metal coil to create electricity? Is there a word for “not greedy”? The antonym of “greedy” is “generous”. Is there a word for “not greedy” (one who is content with what he has)? I would like to know how to find a set of coordinates on a small circle some distance from another point.   For example, lets say A = (39.73, -104.98) #Denver  B = (39.83, -106.06) #point approximately 50 nautical miles away from A along the great circle track to C  C = (40.75, -111.88) #Salt Lake City, approximately 323 nautical miles away from A  How do I get the set of orange coordinates when I don't know the angle but I do know the cross track distance?  For concreteness, lets say the outer points near B have a cross track distance of +- 30 nautical miles and the inner points have a cross track of +- 15nm.  And the points near C have a cross track of +- 25 nm.  Also I have an initial bearing from A to C of -1.34 #radians I know that I can append &amp; to a command to run the process in the background.   I'm SSH'ing into an Ubuntu 12.04 box and running a python program with $python program.py &amp; -- but when I go to close the terminal window I get a message saying that closing the terminal will kill the running process.   Why is this? I am using the ampersand to run the process in the background. How can I get it to run regardless of if I am SSH'ed in? How to return only the Date from a SQL Server DateTime datatype SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that? As figured out in  localized SE sites must not be displayed in the HNQ's. But  actually is.        Has to be fixed I guess? I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses and sheer volume of education into the physical tools and mindset that the other students who took the course had, like Noether's theorem, Lagrangian and Hamiltonian mechanics, statistical methods, and so on.  The courses themselves went well enough. My mathematical experience more or less made up for a lacking physical understanding. However, I still haven't found an elementary explanation of gauge invariance (if there is such a thing). I am aware of some examples, like how the magnetic potential is unique only up to a (time-)constant gradient. I also came across it in linearised general relativity, where there are several different perturbations to the spacetime metric that give the same observable dynamics.  However, to really understand what's going on, I like to have simpler examples. Unfortunately, I haven't been able to find any. I guess, since "gauge invariance" is such a frightening phrase, no one use that word when writing to a high school student.  So, my (very simple) question is: In many high school physics calculations, you measure or calculate time, distance, potential energy, temperature, and other quantities. These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above? Or are these two different concepts? Probability of continuous random variable Using QGIS 3 graphical modeler I created a model with a variety of input parameters (> 10). When starting the model, a nice user interface is generated. But the arrangement of the GUI elements is totally confused and apparently unpredictable during creating the model. It does neither seem to depend on the arrangement of input parameters in the graphical modeler nor on the chronological order in which the input parameters are added to the model.   Is there a way that let's me arrange the GUI elements in the order I desire? Where can I find good literature for beginners? I am trying to find some database tutorials,books and similar, since I have started this course in college, and received some material, but it is really hard to manage through all the info on the net. I searched through the questions on the site but didn't find any. Thankfull for all your input. Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction Prove that $2(\sqrt{n+1}-\sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1})$ if $n \ge 1$ using induction.  Can someone help me with this problem please. Base case is easily shown, and for the inductive step I know I have to use the inequality for $n\gt1$ to show that it is true for $n+1$, but I am not quite sure how to.   Thanks. How can I disable the drag handle on the desktop? Let $$S=\sum_{n=1}^{\infty}x^{n^2}, \quad|x|&lt;1.$$  It is convergent and the sum is definitely less than $1/(1-x).$   Is there any closed form for $S$ ? Finding UMVUE of $\theta$ when the underlying distribution is exponential distribution Hi I'm solving some exercise problems in my text : "A Course in Mathematical Statistics".  I'm in the chapter "Point estimation" now, and I want to find a UMVUE of $\theta$ where $X_1 ,...,X_n$ are i.i.d random variables with the p.d.f $f(x; \theta)=\theta e^{-\theta x}, x\gt0$.   I know that $E(X_i)=1/\theta,$ for each $i$, and also have that $\bar{X}$ (or equivalently $\sum_1^n X_i$) is a complete sufficient statistic for $\theta$. But I cannot go any further here. Somebody can help me? Parse HTML in Android 2018 monthly product team updates Mid-2017  started publishing monthly updates. This post will continue that effort into 2018. (.)  Updates are posted at the beginning of the month. They cover what was done last month, what is still in progress and new work that will be started in the current month.  Updates:      (CoC, nav, OpenID, responsive design, bugs, campaigns)  (CoC, comment flags, nav, OpenID)  (question wizard, nav, user survey, OpenID)  (tag watching, GDPR)  (responsive design, tags, teams)  (responsive design, OpenID, teams, sponsorship)  (responsive design, facebook auth, question lists, teams)  (newsletter, sign up, Documentation reputation, channels)  (top bar issues, hats, question template) When I visit  I see this:     The "Congratulations! You've earned (badge)" section has dropped down completely beneath the tabs under it. The "You've earned a new privilege!" section has a gap at the bottom:     Shog9 says in comments how to reproduce this:     Gotta have earned a badge, not dismissed the notification, and be looking at your own profile.