I am looking for references to the following problem, which I saw a long time ago and I think is a well-known problem (maybe from IMO or American Mathematical Monthly), I hope to remember it correctly.  Problem. Let $a$ and $b$ be two positive integers. If $a^n - 1$ divides $b^n - 1$ for all the positive integers $n$, then $b = a^k$ for some positive integer $k$.  Thank you in advance for any answer.  EDIT: As noted by cr001, a solution to the problem was given on Math.StackExchange (see ). However, what I am asking for is a reference to a journal or book.
We can prove that there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1$. This leads to the following question:     Is it true that for every pair of primes $p&lt;q$ there are only finitely many integers $n$ such that $p^n-1\mid q^n-1$? Are there two primes $p&lt;q$ and an integer $n&gt;p+q$ such that $p^n-1\mid q^n-1$?Is it true that if $n&gt;6$ then $2^n-1\nmid 5^n-1$?   Edit: Here are some examples: \begin{array}{ll} 2^{36}-1\mid 41^{36}-1, &amp;3^{12}-1\mid 97^{12}-1,\\ 5^{6}-1\mid 37^{6}-1, &amp;7^{4}-1\mid 151^{4}-1. \end{array}  Now I prove there is no integer $n&gt;1$ such that $2^n-1\mid 3^n-1.$  Proof: Denote $A=2^n-1$ and $B=3^n-1$. If $n$ is even then $3$ divides $A$ but not $B$, a contradiction.  If $n$ is odd then $A\equiv -5\pmod {12}$. Since every prime greater than $3$ is $\equiv \pm1,\pm5 \pmod {12}$, some prime factor $p$ of $A$ must be congruent to $\pm5 \pmod {12}$. As $p \mid B$, we have $3^{n+1}\equiv 3 \pmod p$. Since $n+1$ is even, we get $(\frac{3}{p})=1$, a contradiction again.
Spotlight overrides first character What's wrong: I hit CMD+SPACE to open up spotlight, typically type in the name of the application I want to start, hit enter and either the wrong app or no app at all will start, because the app name is missing the first character. "Learn to type" I hear you say but wait: as I enter the first character in the search bar, the cursor jumps to the beginning of the line and my second character input overrides the first one. So entering for example "ever", in order to start Evernote, will result in "ver". "php" to start PHPStorm will result in "hp" and start my damn HP printer app and so on.   This cursor jump to the beginning of the line will not happen in any other application and even if I accept the fact that my cursor jumps from time to time, I don't get why the second character overrides the first one. I'm not even sure if you can configure your Mac to always override the character at current position, but I surely haven't.  So this keeps on happening quite regularly and I cannot reproduce it. It just happens and when I try it again (close Spotlight, CMD+SHIFT again), it won't happen again, even if I give it like 5 tries or so.  It won't kill me but I am a software developer myself and this kind of bugs just leave me frustrated plus I can not imagine that this is the desired behavior. I have seen some talk about this issue on the internet but I did not come across a solution..  I have 3 MacBooks, different ages, all with German keyboard layouts and German set as system language and they all do this. Just mentioning this, because after that fact, I can barely imagine this isn't affecting at least a lot, if not all, MacBook (Pro?) users.   13" Pro 2010 or so 15" Pro Retina 2013 13" Pro Retina 2015   They all run El Capitan but I already encountered this before. I don't know the OS X version I had installed when I first stumbled across it. I had Windows 8 installed on my system, which I screwed up, and now I run Ubuntu 12.04. While installing, Ubuntu used all the 461 GB of HDD space (one partition).   Now I want to make a new partition of maybe 100 GB or so, in order to install Windows 7 alongside Ubuntu (with a dual-boot). But I can't even get a partition done. I can't change the partition in GParted either - the option is disabled. Maya is sister of dhara who is doctor. Maya is sister of dhara, who is doctor.   In both statements who is used for which person??? How was Hedwig able to find Sirius? In the beginning of Goblet of Fire, Harry sends Snuffles a letter.  Hedwig somehow finds him after the extremely specific command from Harry, "Just find him, okay?"   Theories:   Hedwig has a scanning system, like Baymax in Big Hero 6. Hedwig got super lucky. Hedwig flies really, really fast.   Now I know that owls are intelligent beings in the Harry Potter canon. But.... those ideas are just as crazy as the next.  So how did Hedwig do it? How to handle calendar TimeZones using Java? I have a Timestamp value that comes from my application. The user can be in any given local TimeZone.  Since this date is used for a WebService that assumes the time given is always in GMT, I have a need to convert the user's parameter from say (EST) to (GMT). Here's the kicker: The user is oblivious to his TZ. He enters the creation date that he wants to send to the WS, so what I need is:  User enters: 5/1/2008 6:12 PM (EST)  The parameter to the WS needs to be: 5/1/2008 6:12 PM (GMT)  I know TimeStamps are always supposed to be in GMT by default, but when sending the parameter, even though I created my Calendar from the TS (which is supposed to be in GMT), the hours are always off unless the user is in GMT. What am I missing?  Timestamp issuedDate = (Timestamp) getACPValue(inputs_, "issuedDate"); Calendar issueDate = convertTimestampToJavaCalendar(issuedDate); ... private static java.util.Calendar convertTimestampToJavaCalendar(Timestamp ts_) {   java.util.Calendar cal = java.util.Calendar.getInstance(       GMT_TIMEZONE, EN_US_LOCALE);   cal.setTimeInMillis(ts_.getTime());   return cal; }   With the previous Code, this is what I get as a result (Short Format for easy reading):  [May 1, 2008 11:12 PM] If a subsequence of a Cauchy sequence converges, then the whole sequence converges. Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon &gt; 0$. Take $N &gt;0$ such that for all $n,m &gt; N$, we have   $$d(x_n,x_m) &lt; \frac{\epsilon}{2}.$$  By hypothesis, we can take also $K &gt;0$ such that for all $n_k &gt; K$, we have  $$ d(x_{n_k},x) &lt; \frac{\epsilon}{2}.$$  Put $M = \max \{N,K\}$. Therefore, for all $n,m,n_k &gt; M$, we have  $$ d(x_n,x) \leq d(x_n, x_{n_k}) + d( x_{n_k},x) &lt; \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$  Hence, $x_n \to x$ as desired.  Is this a correct approach? Thank you very much in advance. Detect init system using the shell Why we won't extract heat from air temperature? Prove continuous functions $f$ and $g$ are equal How can I create a new extensible symbol? Given an arbitrary thin vertical symbol, can I create a new math delimiter from it (and if so, how)?   Edit: Ideally, I would like to be able to create an extensible symbol, either built out an existing normal symbol, or even made from scratch.  Edit 2: I am pretty sure that what I am after is an explanation of how to create a .tfm file for an extensible symbol (which Werner's answer describes the general structure of, but not how to make one).    For example, let's say I wanted to use the \dagger () or \wr () symbol as a delimiter, and write expressions like    \[\left\ldagger \sum_{n=1}^\infty a_n \right\rdagger\]   where the symbol scaled appropriately. Would there be any distinction between declaring a symmetric delimiter (like \vert), as opposed to one that has inherently different "left" and "right" versions like [ and ]?  Now, even being unaware of the inner workings of TeX, I imagine that there has to be lots of information in the fonts for how a delimiter is to be scaled in different situations, and if I try to make my own delimiter, this information won't be present. Would I need to get deep into FontForge or equivalent to achieve my custom delimiters, or is there an easier way? In The Last Jedi:     The Resistance fleet is tracked through hyperspace by the First Order after they make their jump away from their old base.   As soon as the FO ships appear, the Resistance immediately come to the (correct) conclusion as to the tech that was used.  Why would they assume the use of this tech, that they have never encountered before, as opposed to the more common technique of     tracking a ship through the use of a homing beacon. Get a "step-by-step" evaluation in Mathematica Is there a way to find comments you made on YouTube? Closure of $\{\frac{1}{n}\}:=M$ In $\mathbb{R}$ with standard topology the closure of the above set $M$ is $M \cup 0$. I fail showing this by purely topological means. I.e. not using the fact that this holds in the reals as a metric space and deduce it for the induced topology.  I assumed that there exist a closed subset $A$ containing $M$ but not 0. Hence, the complement of $A$ is an open subset containing 0. Does this already yield a contradiction or do I need a different approach?  Could you help me on this? Alka Seltzer Model: formulas/models requested This evening I became fascinated with how my Alka Seltzer tablet disintegrates over time within a small portion of Diet Lipton Citrus Ice Tea. I used a nearly frozen cup; tall, as one might request in a middle-class restaurant.  It is interesting to notice that the Alka Seltzer tablet attempts to float to the top of the glass; but it just cant quite make it, e.g., it keeps bobbing up and down and finally it becomes complete residue.  What are the formulas for this, completely? What might the mathematical model look like from start to finish? My first identifications include, but certainly not limited to: measuring the cup, the milligrams deposited into the cup and of course a the makings of a typical tablet of Alka Seltzer excluding room temperature? Clarify an idiom in a political text Please carefully read the text below:     On 24 November, 1993, a meeting of Leftist intellectuals occurred in   London under the auspices of the Institute for Public Policy   Research (IPPR), which is a Labour-leaning think-tank.  A short document   was circulated in advance of the said meeting to clarify its   purpose.        Among other things, the document declared that the task of the   IPPR was: "To do what the Right did in the seventies, namely to break   through the prevailing parameters of debate and offer a new   perspective on contemporary British politics."    The explanatory document also said, "Our concern is not to engage in a   philosophical debate about foundations of socialism."      If this meant that those foundations were not the appropriate thing to   talk about at the 24 November meeting, then that might have been   right; not every thing has to be discussed at every meeting.  But if   what was meant was that discussion of philosophical foundations is not   what the Left now needs, then I disagree, and if that indeed is what   was meant, then it is curious that the breakthrough by the Right should   have been invoked as an achievement for the left to emulate, for if   there is a lesson for the Left in the Right's breakthrough, it is that   the Left must repossess itself of its traditional foundations on   pain of continuing along its present, politically feeble, reactive   course.  If the Left turns its back on its foundations, it will be   unable to make statements that are truly its own.   Since I've heard only one definition of the part identified in bold, I am at a dead end.  As you know, and according to Oxford's dictionary, the verb "repossess" means: To take back property or goods from SB who has arranged to buy them and can not pay.  Here, the object of verb is a thing not a person, but in the text, first,itself comes before of and its traditional foundations. As a result, I thought that this phrase might be an idiom and so has a different meaning with the original verb.  Please explain the meaning of this phrase. Why does inertia happen? In  R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true? Making ArcPy interactive? Make window always on top? How to remove app which is greyed out, can't be deleted, won't open and doesn't appear in apps list? I have this app with a greyed out icon which I'm trying to delete.    I tap and hold, and it asks me if I want to delete it and its data. I confirm and nothing happens.  I try to open it and nothing happens.  I go to Settings>General>Usage and it's not in the list.  I reboot the device and it's still here.  Is there another way to force deletion of this (ghost) app?