I would like to prove the following equality: $$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}$$if $\gcd(l,k)=1$. What methods can I use? Thank you for your help.
If $p$ and $q$ are distinct primes , prove that $$p^{q-1}+q^{p-1} \equiv 1 \pmod {pq}$$     Using Fermat's Theorem we can get $$p^{q-1} \equiv 1 \pmod q, \qquad q^{p-1} \equiv 1 \pmod p.$$  After that I have no idea at all. Can anyone help me please
Does the sequence $\{\sin^n(n)\}$ converge? Does the sequence $\{\sin^n(n)\}$ converge?   Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge? What are WPF Preview Events? Who took care of Naruto when he was young? In episode 350, we see that the Third Hokage took on to take care of Naruto. But towards the beginning of the anime Naruto is seen to be living alone.   Did Naruto live with the Third? If so, why did he leave there? dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with different file `/usr/lib/xorg/protocol-precise.txt', not allowed The energy of a moving object is $E = mv^2\;.$ That is it increases with velocity squared.  I walk at say 3 miles per hour, or lets round that down to 1 meter per second for a slow walk. I weigh less than $100~\mathrm{kg}\;,$ but lets just round that up to $100~\mathrm{ kg}$ for convenience (it is just after Christmas).  So when I walk along the pavement, I have $100~\mathrm{kg\; m^2 s^{-2}}$, 100 joules of kinetic energy.  Now I get on a passenger jet, which is cruising at around 500 knots, call that 250 meters per second.  In my seat I have $100\times 250^2 = 6250000$ joules of kinetic energy. But when I walk down the aisle I have $100\times 251^2 = 6300100$ joules of kinetic energy. The difference between these is: 50100 joules.  It feels the same to me, walking down the pavement as walking down the aisle of the plane. I didn't have to make some huge effort to get up to speed on the plane, yet I needed 500 times the energy to do it.  How is this possible, and where did the energy come from? Let $f : \mathbb R^n \rightarrow \mathbb R$ be a measurable function.  We know that $f$ is not necessarily locally integrable - but how hard can a measurable $f$ fail to be so? For example, is it true that $f$ is locally integrable outside of a negligible set $N \subset \mathbb R^n$?   Edit: In consideration of the comment by zhs, let me correct the phrasing of the question. I was originally wondering whether the 'singular' behaviour of a measurable function can be localized to a 'small' set.   A motivational example is $f(x) = 1/x$ over the real line. It is locally integrable in the complement of the origin. Hence, a conceivable phenomenon, which I had in mind originally when asking the question, could be that a measurable $f$ is integrable over open sets that are compactly contained in the complement of some negligible sets. Does Stack Exchange contain any edit suggestion or bug submission system, or such type of discussion forum to inform SO or its administrators about suggestions and bug reports that relate to SO itself? Need hint: Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent. Determine whether the series $\sum_{n=1}^{\infty}(\frac{n}{n+1})^{n^2}$ is convergent or divergent.  I've tried applying all the basic tests to no avail. I need to find out what the "trick" for this particular series is. Is there a case where including the same header twice is actually helpful? I'm using the listings package with this \begin command:  \begin{lstlisting}[caption=blabla, label=amb]   This yields a nice code listing with a caption like "Listing 1.1 blabla". Is there a way to change the caption and have something like "Code 1.1 blabla"?  I saw the title command but it doesn't use a numbering system. Any hint will be greatly appreciated! Do I need to use a comma between two adjectives? You can also contact me to customize your template, or create a new unique one.   Do I need to put a comma between the new and unique adjectives in this sentence?  In my native language (if I am not mistaken) we don't put a comma if the adjectives are of a different type. Not sure about English though, and if it's the case here anyway. Any help would be appreciated! state that "at best" is an idiom. But, what is the grammatical function of "at best" (for example, in the below sentences?)        Their response to the proposal was, at best, cool.   The government's response seems to have been at best confused and at   worst dishonest.   If he drops the course now, at best he’ll get an incomplete, and he could fail.      Thanks What are best practices for managing SSH keys in a team? I work with small teams (&lt;10) of developers and admins with the following characteristics:   Most members of the team have >1 personal computer, most of which are portable Team members have access to 10-50 servers, usually with sudo   I think this is pretty typical for most startups and small- to mid-size corporate IT groups.  What are the best practices for managing SSH keys in a team like this?  Should you share a single key for the whole team?  Should each person have their own key on a shared account ("ubuntu" on every server)?  Separate accounts?  Should each team member keep a separate key for each of their laptop or desktop computers? Write access without read access Reprojecting MODIS Swath data to WGS84 using GDAL Trying to run , embedded Tomcat instance fails.  Error seems to appear here:    Apr 17, 2015 2:13:47 AM org.apache.catalina.startup.HostConfig deployDescriptor  INFO: Deploying configuration descriptor /opt/OpenForisCalc/calc-server/tomcat/conf/Catalina/localhost/calc.xml  Apr 17, 2015 2:13:58 AM org.apache.catalina.core.StandardContext startInternal   SEVERE: Error listenerStart   Then if messes up with the rest (this repeats a lot):  Apr 17, 2015 2:13:58 AM org.apache.catalina.core.StandardContext startInternal  SEVERE: Context [/calc] startup failed due to previous errors Interpretation of simple predictions to odds ratios in logistic regression I'm somewhat new to using logistic regression, and a bit confused by a discrepancy between  my interpretations of the following values which I thought would be the same:    exponentiated beta values  predicted probability of the outcome using beta values.    Here is a simplified version of the model I am using, where undernutrition and insurance are both binary, and wealth is continuous:  Under.Nutrition ~ insurance + wealth   My (actual) model returns an exponentiated beta value of .8 for insurance, which I would interpret as:     "The probability of being undernourished for an insured individual is .8 times the probability of being undernourished for an uninsured individual."   However, when I calculate the difference in probabilities for individuals by putting in values of 0 and 1 into the insurance variable and the mean value for wealth, the difference in undernutrition is only .04. That is calculated as follows:   Probability Undernourished = exp(β0 + β1*Insurance + β2*Wealth) /                              (1+exp(β0 + β1*Insurance + β2*wealth))   I would really appreciate it if someone could explain why these values are different, and what a better interpretation (particularly for the second value) might be.     Further Clarification Edits As I understand it, the probability of being under-nourished for an uninsured person (where B1 corresponds to insurance) is:  Prob(Unins) = exp(β0 + β1*0 + β2*Wealth) /               (1+exp(β0 + β1*0+ β2*wealth))   While the Probability of being under-nourished for an insured person is:  Prob(Ins)= exp(β0 + β1*1 + β2*Wealth) /            (1+exp(β0 + β1*1+ β2*wealth))   The odds of being undernourished for an uninsured person compared to an insured person is:  exp(B1)   Is there a way to translate between these values (mathematically)? I'm still a bit confused by this equation (where I should probably be a different value on the RHS):  Prob(Ins) - Prob(Unins) != exp(B)   In layman's terms, the question is why doesn't insuring an individual change their probability of being under-nourished as much as the odds ratio indicates it does?  In my data, Prob(Ins) - Prob(Unins) = .04, where the exponentiated beta value is .8 (so why is the difference not .2?) How can I copy coverart from flac to opus with ffmpeg I am trying to convert some flac files which all have coverart embedded to opus with ffmpeg. All but the coverart gets copied over just fine, but I find myself unable to move the cover over as well. Tried all sorts of -mapmetadata stuff I could find so far, but no luck, the result is always missing the coverart.  While the original .flac reports to have a video stream with the coverart  Duration: 00:02:53.51, start: 0.000000, bitrate: 910 kb/s    Stream #0:0: Audio: flac, 44100 Hz, stereo, s16    Stream #0:1: Video: mjpeg, yuvj420p(pc, bt470bg/unknown/unknown), 500x500 [SAR 1:1 DAR 1:1], 90k tbr, 90k tbn, 90k tbc   the created opus files does not have any graphics:  size=    2815kB time=00:02:53.51 bitrate= 132.9kbits/s speed= 104x video:0kB audio:2793kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.811556%   I tried -vcodec copy, -mapmetadata 0, -mapmetadata 0:s:0 so far, with no luck. Doesnt seem to have anything todo with opus, as I have the same problem converting to aac.  So is there a way to get the cover out of the flac file and move it over to the new file in on go using ffmpeg, or do I have to extract it first and then glue it back again later to the new file with extra commands? Changing the resistor to be in front or behind an LED doesn't affect brightness? What WordPress API function lists active/inactive plugins?