Show that if $p>2 $ is prime, then $8$ divides $p^2 -1$.
For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
whether they are complete metric space I'm quite new to neural networks and currently I'm trying to train a non convolutional neural network on the MNIST data set. I'm observing some behaviour I don't quite understand.   This is the code written with keras as the library:  # the data, split between train and test sets (x_train, y_train), (x_test, y_test) = fashion_mnist.load_data()  y_train = keras.utils.to_categorical(y_train, 10) y_test = keras.utils.to_categorical(y_test, 10) model = Sequential()  early_stopper = EarlyStopping(patience=3)  model.add(Flatten(input_shape=(28,28))) model.add(Dense(128, activation="relu")) model.add(Dense(128, activation="relu")) model.add(Dense(10, activation="softmax")) model.compile(loss="categorical_crossentropy", optimizer="adam", metrics=["accuracy"])  model.fit(x_train, y_train, epochs=20, validation_data=(x_test, y_test), callbacks=[early_stopper])   This gives me a validation_acc of around 20%. The funny thing is, when I change the activation functions to "sigmoid", the loss function to "mean_squared_error" and the optimizer to "sgd" my performance improves to around 85% after 50 epochs.   Having read  I wonder what's the reason for the bad performance of the network I presented in the code. ReLU, cross-entropy and a dynamic optimizer like Adam all seem to improve on the idea of a very vanilla neural network with stochastic gradient optimization, mean squared error as loss and sigmoid activation functions. Yet I get a really bad performance and if I increase the number of nodes in the hidden layers I often get a network that doesn't learn at all.   EDIT: I figured out it has something to do with me not normalizing the input to values between 0 and 1 ... but why is this the problem? Is there a way to prevent those pesky .DS_Store files from being ever created? Is there a way (eg. a binary hack into Finder) to stop it from creating .DS-Stores on LOCAL Volumes?   I am using Leopard, therefore these don't work:    (requires Lion)  (requires Snow Leopard)   Also,  slows down my computer. verifying an incorrect integrating factor I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:  $-y^2dx +x^2dy = 0$  He then asks us to verify that:  $ u(x, y) = \frac{1}{(x-y)^2}$  is an integrating factor.   I multiplied through to get:  $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$   However, the partial derivatives of these do not equal each other so I am a bit confused... How to use both binary and continuous variables together in clustering? I need to use binary variables (values 0 &amp; 1) in k-means. But k-means only works with continuous variables. I know some people still use these binary variables in k-means ignoring the fact that k-means is only designed for continuous variables. This is unacceptable to me.  Questions:     So what is the statistically / mathematically correct way of using binary variables in k-means / hierarchical clustering? How to implement the solution in SAS / R? Solvable Lie algebra with codimension 1 ideal There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"? Why does the Moon not revolve around the Sun directly? Check whether $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$ converges or NOT? Can I buy (liquid) duty free at Dublin airport, before going through pre-clearance? Following on from my , I've been watching the  from the Dublin airport website. That mentions that after checkin, passengers clear normal Irish security checks, then proceed (via optional shopping) to a TSA-style security check before reaching the CBP juxtaposed border control. It also mentions that duty free shopping is available after immigration and customs.  That leads me to wonder - can I buy duty free liquids in the main bit of Dublin Airport, before reaching the US pre-clearance area? Or must all duty free liquids (eg alcohol) be purchased in the smaller duty-free shop after US pre-clearance just before boarding? The tennis statistics at the end of a set always show the number of winners and unforced errors of the players. But when does a point count as a winner and when as unforced error? This has puzzled me for years: How does Blender decide the window where it will display the rendered image when I hit F12 to render? Travelling as unmarried couple in Dubai I was trying to solve the following:  Given: $0 &lt; a &lt; b$ and $n&gt;1$, prove: $$na^{n-1}(b-a) &lt; b^n-a^n &lt; nb^{n-1}(b-a)$$  I managed to get this far using the mean value theorem: $$a^n(b-a)&lt;b^n -a^n&lt;b^n (b-a)$$ Any idea how to continue? Torus object stretched in array curve modifier I have a chain object. And i have a curve it follows. I got it to work for following curve, but chain object is not circles but both parts are more like stretched circles (ovals). Object i made with 2 toruses one rotated 90 degrees and then joined them together. I tried single torus too. But same issue. on a curve it just looks stretched oval. What can be the issue?   Thanks How to add percent sign to NSString Disable sleep on laptop lid close in tty1 I'm using Ubuntu Desktop 15.10 on a laptop and it runs an Apache server with SSH as well. When using a GUI, I have the power options set to not sleep at all even when the laptop lid is closed, but recently I stopped the X-System and am now using TTY1 instead to save resources. However, in TTY1, if I close the laptop lid, the computer goes to sleep and prevents the SSH and Apache server from working. Any way to prevent this easily? The realistic answer I bet is to use Ubuntu Server instead (unless I'm mistaken and Ubuntu Server will also sleep on lid close) but I don't exactly want to switch yet unless there's an easy way to switch to it like you can switch from ubuntu-desktop to lubuntu-desktop? Any advice is welcome to prevent the computer from sleeping How is consensus determined on Meta sites? On our main sites &quot;the best answer&quot; is  as the superposition of the highest voted answer and the accepted answer.  Truth be told, the system is rather insistent on allowing multiple &quot;right answers&quot; and that's one of its strengths: you can be the judge of who you believe. But that doesn't always translate well on the per-site metas, which serve as a sort of policy-creation body.  People who are active on meta help decide how policies are enforced on the main site.  Moderators who have ♦-powers tend to take mandates on meta very seriously.  , accepted answers to meta-questions are no different than any other answer.  Does that leave just votes to determine consensus? Closely related: Not everyone participates on meta-sites.  It can be hard to tell if enough people have looked and voted on questions for real consensus to be declared.  Size of the site determines how many people could be voting on questions.  What counts as a quorum? Finally, sometimes people change their minds about a meta-question, but, , it's not really possible to change one's mind without a pointless edit to someone else's answer.  When should meta-questions be revisited?  On graduated sites, ♦ moderators are elected, which makes them representatives of the active participants, but on beta sites, we are appointed.  It's doubly pressing for beta sites to determine &quot;the will of the people&quot; since part of the point of a beta is to get a working policy framework in place before electing permanent moderators. I'm interested in figuring out how sites already answer these questions and what might be done network-wide to make discovering consensus a little bit easier.  N.B.: Obviously this assumes that consensus matters.  It does to me when I'm making my decisions and I suspect it matters to others.  But that isn't to say that it always makes a difference.  I'm really not interested in answers that hope to use this question as a grindstone for all the axes stored up against certain people who have made unpopular decisions and have the power to make those decisions stick.  Feel free to ask your own question(s).  (Thank you.) finding rational roots of polynomials Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.  I've been looking at some examples, but I get confused. What is your Instagram password when signing in using Facebook? Do this:   On your phone, create an instagram account using Facebook log in. Now on your desktop web browser, log in using Facebook connect.  Now go to delete your account - edit profile -> I'd like to delete my account.    It will ask for your password.   But you never gave it a password in order to create the account.   I tried using my Facebook password, but that didn't work.   Any ideas? The Instagram account management seems quite broken. I know that we don't have automatic login for the Data Explorer, but what's with the automatic timeout/logout?  There have been times that I am logged into the site, maybe I walk away for a little bit, and I come back to being logged out.    Is there a reason for logging us out of Data Explorer?  If so, is it possible to either remove the timeout or extend the period of time before timeout?