Show that if n divides a single Fibonacci number, then it will divide infinitely many Fibonacci numbers.
While studying the Fibonacci sequence I encountered this problem in the handout, and I can not understand how to do it.     Show that if the Fibonacci sequence has a term divisible by a natural number $m$, then there are infinitely many such terms.
Show that $PF.PG=b^2$ in a hyperbola If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then show that $$PF.PG=b^2\space and \space PF.PG'=a^2.$$ We know that the equation of the normal at parametric point $P\equiv(a\sec\theta,b\tan\theta)$ is given by  $$a\,x\cos\theta+b\,y\cot\theta=a^2+b^2$$   Equation of the transverse axis is $y=0$ and that of the conjugate axis is $x=0$.   Hence   $$G\equiv(\frac {a^2+b^2}{a}\sec\theta,0), \, G'\equiv(0,\frac {a^2+b^2}{b}\tan\theta)$$  C is the origin and CF is perpendicular to the normal. Hence   $$ CF\equiv b\,x\sec\theta-a\,y\tan\theta=0\implies y=\frac {b\,x\sec\theta}{a \tan\theta}$$   Substituting this in the equation of the normal we get   $$a\,x\cos\theta+b \,\cot\theta\,\frac {b\,x\sec\theta}{a\,\tan\theta}=a^2+b^2\ $$   implies  $$ (a^2 \cos\theta \sin^2\theta+ b^2\cos\theta)x=(a^2+b^2)\sin^2\theta$$  After finding the coordinates of F, the calculation becomes very complicated. So is there an easier way to approach the problem? I would love to see a pure geometric solution to this problem. I imported the coordinates from an Excel table. They show as single points as intended. Now I want to add columns to the attribute table, but I'm not able to edit it. I wanted to import the following columns from Excel as well, but that doesn't seem to work either.  Sorry for the noob-question, but I couldn't find anything relating the theme through google and such.  Ok, with pictures then: I want to use that to add columns with information to these       You see, the small pensil thingy is greyed out, so I can't edit anything in this layer and I can't add any information to my points.  If I try Ale suggestion I get this error: How do gas molecules constantly bounce without losing energy? From a related question (), i asked myself this: How do gas molecules constantly bounce off each other without losing energy?  If you drop a ball, it bounces a bunch of times, but the height of each bounce gets shorter and shorter. Because it loses energy in each bounce.  So I don't see how a pressurized gas doesn't lose energy. As I understand it, pressure is a manifestation of lots of molecules bouncing off each other. So over time, shouldn't the pressure drop as the molecules lose speed? and eventually all the molecules will settle in a pile on the floor.  I think another way to phrase this is, how do elastic collisions not lose any energy in the exchange? My understanding of the 2nd law of thermodynamics is that some energy is always "lost" when it's converted from one form to another, or transferred from one object to another. I.e., no transfer/conversion of energy is ever 100% efficient. Number of integer solutions of $x^2 + y^2 = k$ I'm looking for some help disproving an answer provided on a  I posted about computing the number of double square combinations for a given integer.    The original question is from the Facebook Hacker Cup       Source: Facebook Hacker Cup   Qualification Round 2011      A double-square number is an integer $X$   which can be expressed as the sum of   two perfect squares. For example, 10   is a double-square because $10 = 3^2 + 1^2$.   Given $X$, how can we determine the   number of ways in which it can be   written as the sum of two squares? For   example, $10$ can only be written as $3^2 + 1^2$   (we don't count $1^2 + 3^2$ as being different). On the other hand,   $25$ can be written as $5^2 + 0^2$ or as   $4^2 + 3^2$.      You need to solve this problem for $0 \leq X \leq 2,147,483,647$.      Examples:      $10 \Rightarrow 1$      $25 \Rightarrow 2$      $3 \Rightarrow 0$      $0 \Rightarrow 1$      $1 \Rightarrow 1$   In response to my original question about optimizing this for F#, I got the following response which I'm unable to confirm solves the given problem correctly.       Source:       Again, the number of integer solutions   of $x^2 + y^2 = k$ is four times the   number of prime divisors of $k$ which   are equal to $1 \bmod 4$.      Knowing this, writing a program which   gives the number of solutions is easy:   compute prime numbers up to $46341$ once   and for all.      Given $k$, compute the prime divisors of   $k$ by using the above list (test up to   $\sqrt{k}$). Count the ones which are   equal to $1 \bmod 4$, and sum. Multiply   answer by $4$.   When I go through this algorithm for $25$, I get $8$ which is not correct.          For each prime factor (pf) of $25$ ($5$, $5$ are prime factors of $25$)         if pf % $4$ = $1$ (true for both $5$'s), add $1$ to count      return $4$ * count (count would be $2$ here).          So for $25$, this would be $8$ UIActionSheet is not showing separator on the last item on iOS 7 GM Can I connect to a sandbox using JWT Flow and Prod connect app? I'm trying to connect to a sandbox using the sfdx cli and the JWT flow through force:auth:jwt:grant. According to , we can use the same connected app information from Production to connect to our Scratch Org.  Is it something that should work for sandbox too ? I'm getting a "Failed: Not approved" error.  Connected app in sandbox comes from the refresh but has a different consumer key. The Permission set is set correctly on my user.  Is there a way to connect to a recently refreshed sandbox using the JWT flow (without manually changing something like the consumer key) ? Why does factoring eliminate a hole in the limit? Can't clean a full /boot because of unmet dependencies I got an error message explaining my /boot is full. trying to clean up old image files always seems to fail because of disk full errors or dependency errors. Could someone explain where I’m going wrong.   $ sudo apt-get autoremove Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run ‘apt-get -f install’ to correct these. The following packages have unmet dependencies.  linux-image-extra-3.13.0-44-generic : Depends: linux-image-3.13.0-44-generic but it is not installed  linux-image-extra-3.13.0-45-generic : Depends: linux-image-3.13.0-45-generic but it is not installed  linux-image-generic : Depends: linux-image-3.13.0-45-generic but it is not installed E: Unmet dependencies. Try using -f.  $ ls /boot abi-3.13.0-32-generic         initrd.img-3.13.0-43-generic abi-3.13.0-36-generic         lost+found abi-3.13.0-37-generic         memtest86+.bin abi-3.13.0-39-generic         memtest86+.elf abi-3.13.0-40-generic         memtest86+_multiboot.bin abi-3.13.0-43-generic         System.map-3.13.0-32-generic config-3.13.0-32-generic      System.map-3.13.0-36-generic config-3.13.0-36-generic      System.map-3.13.0-37-generic config-3.13.0-37-generic      System.map-3.13.0-39-generic config-3.13.0-39-generic      System.map-3.13.0-40-generic config-3.13.0-40-generic      System.map-3.13.0-43-generic config-3.13.0-43-generic      vmlinuz-3.13.0-32-generic grub                          vmlinuz-3.13.0-36-generic initrd.img-3.13.0-32-generic  vmlinuz-3.13.0-37-generic initrd.img-3.13.0-36-generic  vmlinuz-3.13.0-39-generic initrd.img-3.13.0-37-generic  vmlinuz-3.13.0-40-generic initrd.img-3.13.0-39-generic  vmlinuz-3.13.0-43-generic initrd.img-3.13.0-40-generic   I used   kernelver=$(uname -r | sed -r 's/-[a-z]+//') dpkg -l linux-{image,headers}-"[0-9]*" | awk '/ii/{print $2}' | grep -ve $kernelver   to get   linux-headers-3.13.0-32 linux-headers-3.13.0-32-generic linux-headers-3.13.0-36 linux-headers-3.13.0-36-generic linux-headers-3.13.0-37 linux-headers-3.13.0-37-generic linux-headers-3.13.0-39 linux-headers-3.13.0-39-generic linux-headers-3.13.0-40 linux-headers-3.13.0-40-generic linux-headers-3.13.0-44 linux-headers-3.13.0-44-generic linux-headers-3.13.0-45 linux-headers-3.13.0-45-generic linux-image-3.13.0-32-generic linux-image-3.13.0-36-generic linux-image-3.13.0-37-generic linux-image-3.13.0-39-generic linux-image-3.13.0-40-generic   Then picking the first image to remove   $ sudo apt-get purge linux-image-3.13.0-32-generic Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies.  linux-image-extra-3.13.0-32-generic : Depends: linux-image-3.13.0-32-generic but it is not going to be installed  linux-image-extra-3.13.0-44-generic : Depends: linux-image-3.13.0-44-generic but it is not going to be installed  linux-image-extra-3.13.0-45-generic : Depends: linux-image-3.13.0-45-generic but it is not going to be installed  linux-image-generic : Depends: linux-image-3.13.0-45-generic but it is not going to be installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution). Was Mjolnir ever used for building something? In the movie Thor (2011), while he was about to crown Thor as the next king of Asgard, Allfather Odin describes the mighty hammer Mjolnir as the following:      So long entrusted with this mighty hammer, Mjolnir. Forged in the heart of a dying star, from the sacred metal of Uru. Only one may ift it. Only one is worthy. Who wields this hammer commands the lightning and the storm. Its power has no equal -- as a weapon, to destroy, or as a tool, to build. It is a fit companion for a King.   We know Mjolnir as a weapon of destruction but are there references where it is used to build something? How to sort a Google Sheets Query by a custom ordering I have a table of choir members with a “name” and a “voice type” column. How can I write a query that orders them by decreasing pitch level of the voice type (i.e. soprano > contralto > tenor > bass)?  Is there in the  something similar to ?  I also tried the , but it does not seem to have an option for custom ordering, either. What does the size of the standard deviation mean? According to  (my emphasis),     In statistics, the standard deviation ... is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.   The purpose of the standard deviation (SD), then, is to tell us how varied or uniform (SD → 0) the data is.  So, given a certain SD, how varied is the data? What does the size of the standard deviation mean?  Please explain the meaning of the SD by interpreting an SD = 1 (M = 0).  If you cannot interpret the size (quantity) of this SD, what other information would you need to be able to interpret it, and how would you interpret it, given that information? Please provide an example.  If, on the other hand, the quantity of the SD cannot be qualified in this manner, my argument is that it is essentially meaningless. If you disagree, please explain the meaning of the SD.  &nbsp;    The following are earlier versions to give context to the answers. Unfortunately these didn't really convey what I wanted, and my attempt to ask it elsewhere was closed. (I don't need these versions answered now):  First revision:  What does the size of the standard deviation mean?  For example, if I want to study human body size and I find that adult human body size has a standard deviation of 2 cm, I would probably infer that adult human body size is very uniform, while a 2 cm standard deviation in the size of mice would mean that mice differ surprisingly much in body size.  Obviously the meaning of the standard deviation is its relation to the mean, and a standard deviation around a tenth of the mean is unremarkable (e.g. for IQ: SD = 0.15 * M).  But what is considered "small" and what is "large", when it comes to the relation between standard deviation and mean? Are there guidelines similar to the ones that Cohen gives for correlations (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?    Original question:  We always calculate and report means and standard deviations. But what does the size of the variance actually mean?  For example, assume we are observing which seat people take in an empty room. If we observe that the majority of people sit close to the window with little variance, we can assume this to mean that people generally prefer siting near the window and getting a view or enough light is the main motivating factor in choosing a seat. If on the other hand we observe that while the largest proportion sit close to the window there is a large variance with other seats taken often also (e.g. many sit close to the door, others sit close to the water dispenser or the newspapers), we might assume that while many people prefer to sit close to the window, there seem to be more factors than light or view that influence choice of seating and differing preferences in different people.  At what values can we say that the behavior we have observed is very varied (different people like to sit in different places)? And when can we infer that behavior is mostly uniform (everyone likes to sit at the window) and the little variation our data shows is mostly a result of random effects or confounding variables (dirt on one chair, the sun having moved and more shade in the back, etc.)?  Are there guidelines for assessing the magnitude of variance in data, similar to Cohen's guidelines for interpreting effect size (a correlation of 0.5 is large, 0.3 is moderate, and 0.1 is small)?  For example, if 90% (or only 30%) of observations fall within one standard deviation from the mean, is that uncommon or completely unremarkable? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. Objects disappear when looking through camera How do I calculate if a spell hits and how much damage it does? I'm not entirely sure I understand what the PHB is telling me about spell attacks.  Say I'm a wizard who's casting Fire Bolt (I could also be a Cleric casting Inflict Wounds or what-have-you), the spell itself doesn't say anything about making an attack roll or the enemy making a saving throw.    So do I make an attack roll and then add in my INT mod + proficiency bonus?  Or does the target have to make a saving throw against a DC of 8+ my INT mod + proficiency bonus?  And, assuming I hit, do I add my INT mod + proficiency bonus to the damage? Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$ This is being asked in an effort to cut down on duplicates, see here: , and here: .    What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$  In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} +\cdots +a_1 x+ a_0$, how would I find $$\lim_{x\rightarrow\infty} P(x)^{1/n}-x.$$  For example, how would I evaluate limits such as $$\lim_{x\rightarrow\infty} \sqrt{x^2 +x+1}-x$$ or $$\lim_{x\rightarrow\infty} \sqrt[5]{x^5 +x^3 +99x+101}-x.$$ Can you transit through the Schengen Area without a visa when traveling on separate tickets? I am planning to travel from US to Albania, and when I search on flight ticket websites such that US as origin and Tirana (TIA) as destination, then it costs me $1050 cheapest. However, it costs me $850 when I buy two separate round-trips through Barcelona (US to Barcelona round-trip, Barcelona to Tirana round-trip). I just checked Barcelona, but there might be other transfer airports in Schengen area which costs me even less. I don't need a visa for Albania but I need a visa to go into a Schengen country, so my question is 3-fold:   How can I learn which Europe airports have the international transfer area, so that I don't need a Schengen visa to transfer through the airport? I guess the airlines would take care of that when you buy the tickets together (I already have such an experience multiple times), but given that I will buy them separately, I need to have that information beforehand. The two separate flights I am planning to buy are through different airlines, and in my experience, the airlines check whether the traveler has a visa for the destination country before they let you into the airplane. So, would it be easy to convince the US to Barcelona trip airline that Barcelona is not my final destination, but I have a separate flight from Barcelona to Albania? Are there any Europe airports that have "international baggage claim" areas which are before the passport checkpoint? I mean, if I buy separate flights, do I have any chance to check-in a baggage for the US to Barcelona trip? How can I block all ports except some? I am trying to use iptables but I am clueless in understanding the documentation even so that I've locked myself out of my server twice already whilst trying to implement it.  These are the simple rules that I want to implement:   Allow ports 80, 6667 and 6697 to everyone Allow port 22 to just my IP for example 10.10.10.10   Please can someone help me implement these simple rules, without locking myself out? How to wrap lines correctly inside algorithmic I am having problems displaying algorithms. The wrapping is really ugly and I can't, for the life of me, find a fix that doesn't involve manually tweaking the hell out of everything, to the point that I might as well scrap the crappy algorithm packages and write everything out in TeX.  Is there any way to properly indent the following  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}    \caption{My pseudo code.} \begin{algorithmic}[1]    \While{this line is sooooooooooo long and boring and too much for algorithmic to handle}       \State look at this state, this state is just too long for algorithmic to display it properly        \While{again this line is sooooooooooo long and boring and too much for algorithmic too handle}          \State look at this state again, this state is just too long for algorithmic to handle, I'm just going to switch to Word       \EndWhile    \EndWhile \end{algorithmic} \end{algorithm} \end{document}​     so that the output is similar to Proving divisibility of expression using induction Daily reputation cap doesn't account for reputation * loss* due to bounty awarded? There are many Q&amp;A's around this and the best explanation of daily cap model is from this answer that uses . It is an old answer but gives a good understanding.   My confusion is in applying this model to my situation yesterday.    On 31/12/2019, I awarded a bounty to , since it became eligible for bounty (2 days after posting it). Subsequently, I got more upvotes (including one accept) which exceeded the daily cap of 200. I am confused about this-loss of reputation due to bounty awarded by me preceeded subsequent gain of reputation so going by the bucket analogy, I should have my rep adjusted accordingly and those 6 up votes (at the bottom of table) should have been credited (rep should have been 177 instead of 117),  /reputation shows this   2 219845 (10)  1 219884 (2)  2 170717 (10)  2 219621 (10)  2 219845 (10) 2 219845 (10) 2 218537 (10)  2 219845 (10)  2 219892 (10)  2 219845 (10) 8 219821 (-100)  2 219845 (10)  2 219621 (10)  2 219845 (10)  2 219821 (10)  2 219845 (10)  2 219845 (10)  2 219845 (10)  2 219912 (10)  1 219912 (15)  2 219845 (10)  2 219845 (10)  2 219892 [10]  2 219845 [0]  2 219621 [0]  2 219845 [0]  2 219845 [0]  2 219845 [0]  2 219845 [0]  -- 2019-12-31 rep +117 = 27089 --   It is not the 6 votes I am bothered about, it is more about trying to correct my understanding  or have I stumbled upon an anomaly?