I'd like to show the following: Let $a \in \mathbb{N}$ and $p \in \mathbb{P}$ with $gcd(a, p-1)=1$, then for every integer $b$, the congruence relation $x^a \equiv b$ mod $p$ has a solution.  If $p \mid b$, then this trivially holds. For the other case, I've started with showing that there exists a $c$ such that $a*c \equiv 1$ (mod $p-1$) and I think that $x = b^c$ should satisfy the relation, but I've not yet been able to show this. Can someone help me?  Thanks!
The question is about the encryption equation in asymmetric encryption.  For $c\equiv m^e\bmod n$, prove that for every unique $m$ there is a unique $c$.  Here $n=pq$ where $p$ and $q$ are primes larger than $m$ and the exponent $e$, is $\geq3$ and coprime with $p-1$ and $q-1$.  Thanks!
Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem) As I have heard people did not trust Euler when he first discovered the formula (solution of the ) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.   I believe many of you know some nice proofs of this, can you please share it with us? How can I record my screen on Ubuntu?   The app I'm looking for has ideally all of these features:   Can record in a format that can be played back easily on any platform and/or accepted by YouTube or another popular video site Can record just a window (instead of the whole screen), possibly selecting it with a mouse click Can start recording after a configurable delay (e.g., I launch the app and have time to do arrangements to my desktop/window before actual recording starts) Using "you" and "your" as a representation for yourself and everyone in general Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. I made a simple FM transmitter following  and it worked! Now I need some help to understand how.  My coil is made of 1 mm copper wire has 6 turns and 8 mm wire diameter. According to this online calculator:  this means that: L = 3.9e-7 H  I did not have any 5 pF trim capacitor so I used a 3 pF capacitor instead.  Am I correct in assuming that the frequency this will transmit on can be found by looking at the L1 C2 as a LC oscillator and using the frequency formulae:  fR = 1/(2*pi*sqrt(LC))  ?  If so I should get fR = 147 MHz. Which is strange since I received the signal on 91.5 MHz on my radio.  In a LC circuit I understand that the signal decays exponentially over time. That is hopefully not happening here. Why? Because of C4 maybe?  Also what is the purpose of C1?  If I want to increase the range of this FM transmitter somewhat can I do this by simply replacing R2 with a 470 Ohm resistor?  Are there free simulator programs that let me see the resulting spectrum at the antenna? The following code receives seg fault on line 2:  char *str = "string"; str[0] = 'z';  // could be also written as *str = 'z' printf("%s\n", str);   While this works perfectly well:  char str[] = "string"; str[0] = 'z'; printf("%s\n", str);   Tested with MSVC and GCC. anyone tell how is this effect created which change lips color to orange and little orange on face without brush etc tool how to change the only lips color to orange without brush etc tool and how this is created and little orange on face How to get access to USB-storage from an application installed as Snap? How do I use Drush 8 site aliases in Drush 9? Favorite Tag Buttons partly sticking off page I appear to have found a bug, when adding Favorite tags, I find that the buttons are half off the page (this is of the  site):    Although on other sites it looks more like this:    I am running Ubuntu GNOME 15.04 with Firefox 41.0.1+build2-0ubuntu0.15.04.2. How can latex and pdflatex be both symbolic links to same executable (pdftex) and not behave the same? When I run  $ latex my_doc.tex   and   $ pdflatex my_doc.tex   it works, and when I run  $ pdftex my_doc.tex   it doesn't work.  The thing is I noticed that both LaTeX and pdflatex commands are symbolic links to the same pdftex executable. That is,  $ which latex pdflatex pdftex   yields  /usr/bin/latex /usr/bin/pdflatex /usr/bin/pdftex   and   $ ls -l /usr/bin/latex /usr/bin/pdflatex /usr/bin/pdftex   yields  /usr/bin/latex -&gt; pdftex /usr/bin/pdflatex -&gt; pdftex /usr/bin/pdftex   So how is it possible that commands being symbolic links to a same executable do not behave in the same way ?  Thanks.  I have Ubuntu 12.10 with TeXLive-full 2012.20120611-4 installed. What's the difference between &lt;div class=""&gt; and &lt;div id=""&gt; when it comes to CSS? Is it alright to use &lt;div id=""&gt;?  I see different developers doing this in both ways, and since I'm self taught, I've never really figured it out. Error java.lang.OutOfMemoryError: GC overhead limit exceeded I have encountered this question while tring to figure out why every sequential space $X$, is a quotient spaces of a metric space. If I understand correctly, given a sequential space, every sequence (including it's limit) can be identified with the space $Y=\{0\} \cup \{\frac1{n+1}|n \in N\}$. So The space $X$ can viewed as a quotient space of $\bigoplus_{(x_n) \in C} \{(x_n) \} \times Y$ where $C$ is the set of all converges sequences in $X$. My question is, How can I show that this space is metrizable?  Thank you! Shir What is the difference between Fact and Truth? I'm curious about the difference between Fact and Truth. I was searching on the internet if I could find it. But still I'm confused about the exact meaning. I first read the forum discussion here  where an author has given two examples for each like below     A fact is a reality that cannot be logically disputed or rejected. If   I say "fire is hot," I don't care how great your reasoning skills are,   if you touch fire your skin will burn (and don't give me that "but   people can walk on hot coals!" bull. There's a difference between the   transfer of heat through conduction and training one's body to deal   with the agonizing pain of said conduction). Now when I say this, I am   not speaking a truth, I am speaking a fact. If you say "fire is not   hot," you are not lying, you are incorrect. Facts are concrete   realities that no amount of reasoning will change. When one   acknowledges a fact, they are doing just that. Facts are not   discovered, facts are not created, facts are simply acknowledged.      A truth on the other hand, is almost the opposite. Truths are those   things that are not simply acknowledged, but must be discovered, or   created. If I say "God exists," and I possess strong reasoning for the   affirmative of that statement, then God really does exist, that is a   reality. However, if another individual possesses strong reasoning for   the negative, and because of this reasoning they believe that God does   not exist, then that is also a reality. If we were to debate our   ideologies, and my reasoning appeared stronger than theirs, they may   choose to adopt my belief that God does exist. If they do, then the   existence of God is just as true as the nonexistence of God which they   believed a week ago. Truths, as opposed to fact, are much more fluid   and malleable than their empirical counterparts.   and followed by further discussion.  Then I found this . Article from above link says like below:     Facts are notes and lyrics on sheet music.  Truth is what the singer   gives to the listener when she’s brave enough to open up and sing from   her heart.   But still curious about the difference between both of them.  In our daily life, in general conversation, we generally use these both terms interchangeably. Then what is the difference? Are they synonym or have specific difference? Speed of neutrinos Everyone knows it is close to $c$, but how close? What are the recent results? I pressed F2 to take a few Minecraft screenshots. Later I couldn't find them. Where are they stored on my computer? What is the meaning of $\exp(\,\cdot\,)$? What is the meaning of the notation $\exp(\text{expression})$ ?  I think that it's something of the form $a^\text{expression}$ but does it mean that the base $a=e$ or can it be any base? What causes the portraits of Wizards to behave interactively? Does J.K. Rowling address how the interactive portraits work? Would a portrait painted after a Wizard/Witch died become interactive, or does the painting of the portrait have to take place during the Wizard/Witches lifetime? Can I update an early Airport Time Capsule? What's your preferred way of getting current system status (current CPU, RAM, free disk space, etc.) in Python? Bonus points for *nix and Windows platforms.  There seems to be a few possible ways of extracting that from my search:   Using a library such as  (that currently seems not actively developed and not supported on multiple platform) or something like  (again no activity since 2007 it seems and no support for Windows). Using platform specific code such as using a os.popen("ps") or similar for the *nix systems and MEMORYSTATUS in ctypes.windll.kernel32 (see ) for the Windows platform. One could put a Python class together with all those code snippets.   It's not that those methods are bad but is there already a well-supported, multi-platform way of doing the same thing?