I need to solve $$3x^4-3x^2+1=y^2$$ over the positive integers.  I've tried setting $x^2=t$ and considering the discriminant. I've tried rewriting into $3(2x^2-1)^2+1=(2y)^2$. I've tried substituting $x=3a\pm 1$. The problem would be easy by bounding if the leading coefficient is a perfect square.
I know $4p^4+1$ can't be square number, cause $4p^4&lt;4p^4+1&lt;4p^4+4p^2+1$ for all natural number p. But I don't know the way to prove $3p^4-3p^2+1$ can(not) be square number. Is there a well known way to prove it?
Odds of rolling doubles or triples given N 6-sided dice Over the holidays I played a dice game where each player had 3 to 7 d6 to roll each turn.  The game gave certain advantages to doubles and triples.  I wanted to know the odds of rolling doubles or triples given N dice to understand the importance of "upgrading" to more dice.  My question is really about how to reason about such problems:  It's easy to see that the odds of rolling doubles with 2 d6 are 1/6 (first die can be anything, and 1/6 of the time the second will match).  It's also easy to see that with 7 dice the odds are 1.0 (6 dice could be 1..6, but the 7th must match one of them).  The middle cases are fuzzier to me.  I ended up writing a program to enumerate all of the possibilities (even 6^7 is only 279,936 cases to evaluate) because that was the only way I felt I could verify any closed formula I came up with.  This also enabled me to distinguish cases of multiple doubles (or triples).  I would like to know how to derive closed forms and how to validate such solutions without brute-force simulation. Does WaitForMultipleObjects() reset all auto-reset events? when is f@g not the same as f[g]? How could I place objects (buildings) on a flat surface in the BGE? I'm working on an  in BGE, and I am not sure how I could make it so you can place buildings on flat surfaces (or unless there is an easier way). I would love to know how to place buildings when you click a certain button to place it.  I don't really need anything fancy like price and stuff like that yet. I really have no idea if  is required to do this. The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)...(1+x^{100})$.  I tried the following concept, how to sum 9 using 1-9 only without repetition.  1:9, 2:1+8,3:7+2, 4: 6+3, 5: 5+4,  6:1+2+6 ,7:1+3+5, 8:2+3+4  The answer is 8.  How will it be solved using Binomial theorem. Modify systemd unit file without altering upstream unit file I have installed the pimd service by means of apt. This comes with an upstream systemd unit file (/lib/systemd/system/pimd.service).  I want the service to be restarted when for some reason it gets killed, hence I wish to add the line Restart = always in the unit file.   However, I don’t want to modify the upstream unit file.  Is there any workaround for this? I am having problems with the su command. I know my password and I am typing it correctly but su indicates authentication failure.  So I checked on the internet and then went into recovery mode and changed my username's password to what I was entering before.  Even now entering the same password on su gives me authentication failure.  Please suggest what am I doing wrong?? Prove $[0, 1]$ and $(0, 1]$ are equinumerous by use of bijection What my thought was that the bijection was a piecewise function:  $f(x) = 1/n+1$ if $x = 1/n$ for some $n \in \Bbb N$ and $x=1/n$ if $x \neq1/n$ for some $n \in \Bbb N$.  However, the textbook says this is incorrect. I don't actually see why. Was it a matter of brackets or the open variable $(0,1]$, which is what I tend to think. How does Lockable Suspension work? I recently asked a question about . There  told me about the “blow off” feature. Now, I am curious about how the whole system works.   So, can anyone tell me about how a lockable front suspension works, together with features like “remote-lock” and “blow-off”. Check if partition is encrypted Reset the password in Ubuntu / Linux Bash in Windows I have installed Linux Bash on Windows 10 (available from Microsoft), but I've forgotten my password.  I've searched for a solution but I only find suggestions that involve doing something before the system boots, like accessing recovery mode, but that will not work in this case as this Ubuntu is just an application within a Windows system and doesn't have its own boot process.  How can I reset my password? Summing the power series $\sum\limits_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}\prod\limits_{k=1}^n\frac{2k-1}{2k} $ I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|x|&lt;1$. Google Maps Location History is showing an incorrect location I have a major problem that I hope someone can help with. On my Google Maps feature under my Location History it shows a location that I was never at at any point, but when my wife happened to see that location she doesn't believe me that the GPS could be wrong and pinpoint that location if I was never there.  I was first wondering myself how could that could be possible if I was never there or my phone was never at that location that was 35 miles away from me and second is there any way I could prove the GPS being wrong? Can I contact Google Maps and can they give an official printout of that certain day and every location I was? I really need to prove my honesty to her and hope there is a way. Stop execution of Javascript function (client side) or tweak it Why we use $L_2$ Space In QM? I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space of state functions, from where this arises? and why 2-order space that assumes the following inner product:  $\langle\phi|\psi\rangle =\int\phi^{*}\psi\,dx$  While there is many ways to define the inner product.  In Physics books, this always assumed as given, never explains it, also I tried to read some abstract math books on this things, and found some concepts like "Metric weight" that will be minimized in such spaces, even so I don't really understand what is behind that, so why $L_2$? what special about them? Who and how physicists understood that those are the one we need to use? Undo apt-get remove open* ubuntu 12.04 An init-script does not get called on shutdown I have set up an init-script to control the state of a VirtualBox VM:  #!/bin/sh #chkconfig: 35 99 5 #description: vTiger virtual machine  ### BEGIN INIT INFO # Provides: vtigervm # Required-Start: $local_fs # Requider-Stop: $stop_fs # Default-Start: 3 5 # Default-Stop: 0 1 2 6 # Short-Description: Manage vTiger virtual machine # Description: Utility to start and stop vTiger virtual machine on VirtualBox ### END INIT INFO  start() {     echo -n "Starting vTiger"     echo     su myuser -c '/usr/bin/VBoxManage startvm "vTiger" --type headless'     echo "Started virtual machine" &gt;&gt; /var/log/messages }  stop() {     echo -n "Shutting vTiger down..."     echo     su myuser-c '/usr/bin/VBoxManage controlvm "vTiger" acpipowerbutton'     while [ ! -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; do         echo -n "."         sleep 1     done     echo "Done."     echo "Stopped virtual machine" &gt;&gt; /var/log/messages }  status() {     echo -n "Running VMs: "     su myuser -c '/usr/bin/VBoxManage list runningvms'     echo     if [ -z "`su - juhani -c '/usr/bin/VBoxManage list runningvms | grep vTiger'`" ]; then         RETVAL=3         else         RETVAL=0     fi     echo "Queried virtual machine status" &gt;&gt; /var/log/messages }  echo "Called virtual machine management script with: $1" &gt;&gt; /var/log/messages case "$1" in   start)         start    ;;    stop)         stop    ;;    restart|try-restart|condrestart|reload)         stop         start    ;;    status)         status    ;;    *)         echo $"Usage: $0 {start|stop|restart|status}"         exit 1    ;; esac  exit $RETVAL   The system is CentOS 6.5. If I manually run service [start|stop|status] vtigervm, it works as I expect it to. If the VM is running, $? after the service vtigervm status returns 0, and 3 if it is stopped. I installed it with chkconfig and it created among a few others rc5.d/S99vtigervm and rc0.d/K05vtigervm.  The problem  When I start the system it starts the "service", but on shutdown it does not even run the scripts.  grep "virtual machine" /var/log/messages shows:  *[machine starting]* Called virtual machine management script with: start Started virtual machine *[shutdown -h now]* *[machine stopped]*   What I expect:  *[machine starting]* Called virtual machine management script with: start Started virtual machine  *[shutdown -h now]* Called virtual machine management script with: status Queried virtual machine status Called virtual machine management script with: stop Stopped virtual machine *[machine stopped]*   File permissions &amp; etc:  # ls -lah /etc/rc0.d/ lrwxrwxrwx. 1 root root 13 7.2. 17:49 /etc/rc0.d/K05atd -&gt; ../initd.d/atd lrwxrwxrwx. 1 root root 18 9.2. 00:06 /etc/rc0.d/K05vtigervm -&gt; ../initd.d/vtigervm Integral $\int\frac{2x^2}{2x\cos(2x)+(x^2-1)\sin(2x)} \mathrm d x$ Integrate  $\displaystyle\int\dfrac{2x^2}{2x\cos(2x)+(x^2-1)\sin(2x)} \mathrm d x$   I tried dividing by $\cos^2(x)$ and then substituting $\tan(x)=t$. A well-known application of the MVT is to prove that any $f: \mathbb{R} \to \mathbb{R}$ with $f'= 0$ is constant. But of course, the MVT relies fundamentally on the properties of $\mathbb{R}$, and in particular does not hold in $\mathbb{Q}$. A standard counter-example is the function $f :\mathbb{Q} \to \mathbb{Q}$ defined by $ f(x)=0 $ if $ x &lt;\sqrt{2}$, and $f(x)=1$ if $ x &gt; \sqrt{2}$, for this function is locally constant. There are several ways of thinking about the difference between $\mathbb{R}$ and $\mathbb{Q}$ : $\mathbb{R}$ is connected while $\mathbb{Q}$ is totally disconnected, closed bounded intervals are compact in $\mathbb{R}$, but not in $\mathbb{Q}$, $\mathbb{R}$ satisfies the least upper bound axiom, $\mathbb{Q}$ does not.   Now, I wanted to see how far one could push this counterexample - is there a function $f :\mathbb{Q} \to \mathbb{Q}$ such that $f'=0$ and $f$ is strictly increasing? Clearly, any slight modification of the usual counterexample where $f$ is locally constant will not do. Also, one can show that if $f$ is uniformly differentiable with derivative zero, it must be constant, even in $\mathbb{Q}$, so I knew I was looking for non-uniformly differentiable but still differentiable $f$.   Eventually, I came up with a construction, albeit not a very explicit one: the idea is that, since $\mathbb{Q}$ is countable, one can enumerate it $q_{1}, q_{2}, q_{3}, \cdots$, and then define $f: \mathbb{Q} \to \mathbb{Q}$ inductively and recursively. If one could define $f$ in such a way that $ \forall \ q_{n} \in \mathbb{Q} \ \exists \ c_{n} \in \mathbb{Q} : \forall \ x \in \mathbb{Q} \ |f(x)-f(q_{n})| \leq c_{n}|x-q_{n}|^2$, and $f$ is strictly increasing, we would be done. The recursive definition essentially defines $f(q_{n+1})$, after having defined $f(q_{1}),f(q_{2}),f(q_{3}), \cdots , f(q_{n})$, in such a way that these inequalities hold and $f$ is strictly increasing, and at each step the new $c_{n}$ is chosen large enough so that these inequalities do not become incompatible at subsequent steps ( I am willing to provide more details of this construction if asked).  Note that, as $n \to \infty$, we must have $c_{n} \to \infty$, so that $f$ is not uniformly differentiable. Interestingly, the only key fact about $\mathbb{Q}$ used in the construction is that it is countable, so in a sense the construction provides a proof that $\mathbb{R}$ is uncountable, given the MVT.  While the construction works, it is not very simple nor explicit, which is perhaps to be expected, and the function must be evaluated point by point in the order given by the enumeration of $\mathbb{Q}$. So I was wondering if anyone else knows or can come up with a simpler construction of a function $ f : \mathbb{Q} \to \mathbb{Q}$ such that $f'=0$ and $f$ is strictly increasing. Any thoughts and ideas are welcome.  Edit: I've asked the same question on MO: Define additional math operators to be typeset in roman Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands?