Question: If $n$ is any positive integer I write $F_0=0, F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. The numbers $F_n$ are called the Fibonacci numbers. Let $a_n=(-1)^{\gcd(n,F_n)}$. Is it true that if $a_n=1$ then $n$ is a multiple of $6$ ?   Not much motivation here just an  experimental observation I noticed. I am not sure where exactly to start since I do not know a formula for $\gcd(n,F_n)$.
The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \&amp; \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$   Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green}{n = 1 }}$ $2 \mid F_1$ is false. Also, $3 \mid 1$ is false. The implication [False $\iff$ False] is vacuously true.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Induction Hypothesis}}}$ Assume that $2 \mid F_i \iff 3 \mid n$ for every integer $i$ with $1 ≤ i ≤ k$.  $\bbox[5px,border:1px solid green]{\color{green}{k + 1 \text{th Case}}} \;$  To prove:  $\quad 2 \mid F_{k + 1} \iff 3 \mid k + 1.$   $\bbox[5px,border:1px solid green]{\color{green}{n = k + 1 = 2}} \;$ $2 \mid F_2$ is false. Also, $3 \mid 2$ is false. So [False $\iff$ False] is vacuously true.  Hence assume that $k + 1 ≥ 3.$ We now consider three cases:  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 1: } k + 1 = 3q}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 \require{cancel}\cancel{\mid} (k − 1)$. By the ind hyp, $3 \require{cancel}\cancel{\mid} k    \iff F_k$ odd &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Since $F_{k+1} = F_k + F_{k−1}$, thus $F_{k+1}$ = odd + odd = even.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 2: } k + 1 = 3q + 1}}$  Thus $3 | k$ and $3 \require{cancel}\cancel{\mid} (k − 1).$ By the ind hyp, $3 | k \iff F_k$ even &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Thus $F_{k+1}$ odd.  $\bbox[5px,border:1px solid green]{\color{green}{{\text{Case 3: }} k + 1 = 3q + 2}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 | (k −1).$ By the ind hyp, $3 \require{cancel}\cancel{\mid} k \iff F_k$ odd and $3 \mid (k − 1) \iff F_{k - 1}$ even. Thus $F_{k+1}$ odd. $\blacksquare$  $\Large{1.}$ Does the proof clinch the $(\Leftarrow)$ of the $(k + 1)$th case?   $\Large{2.}$ Since the recursion contains $n, n - 1, n - 2$, thus the recursion "time lag" is $3$ here. So shouldn't $3$ base cases be checked?   $\Large{3.}$ Further to #2, shouldn't "assume $k + 1 \geq \cancel{3} 4$" instead?   $\Large{4.}$ Shouldn't the $n = k + 1 = 2$ case precede the induction hypothesis?  I referenced . Source: Exercise 6.35, P152 of Mathematical Proofs, 2nd ed. by Chartrand et al    Supplement to peterwhy's Answer:  $\Large{1.1.}$ I wrongly believed that all 3 Cases proved the $\Leftarrow$. I now see that Case 1 is $\Leftarrow$ via a Direct Proof. Cases 2 and 3 are $\Rightarrow$ via a Proof by Contraposition. Nonetheless, how would one foreknow/prevision to start from $3 \mid n$  for both directions of the proof?
Page range truncation with imakeidx This is a followup question to egreg's answer to  The answer runs into a problem if an index topic occurs in more than one page range. The following MWE, based on egreg's answer:  \begin{filecontents*}{subjind.ist} delim_0 ", \\checkrange{" delim_t "}" \end{filecontents*}  \documentclass{book} \usepackage{xparse} \usepackage{imakeidx}  \makeindex[title=Subject Index,options=-s subjind]  \ExplSyntaxOn \NewDocumentCommand{\checkrange}{&gt;{\SplitArgument{1}{--}}m}  {   \formatrange#1  } \NewDocumentCommand{\formatrange}{mm}  {   \IfNoValueTF { #2 }    { % no --, it's not a range     #1    }    { % it's a range     \egreg_range:nn { #1 } { #2 }    }  }  \bool_new:N \l__egreg_range_compare_bool  \cs_new_protected:Nn \egreg_range:nn  {   % print the start number   #1--   \int_compare:nNnTF { \str_count:n { #1 } } = { \str_count:n { #2 } }    {% same number of digits, remove equal ones at the start     \__egreg_range_compare:nn { #1 } { #2 }    }    {% different number of digits, print both     #2    }  } \cs_new_protected:Nn \__egreg_range_compare:nn  {   \bool_set_true:N \l__egreg_range_compare_bool   % do a loop on the digits in the first number   \int_step_inline:nn { \str_count:n { #1 } }    {     \bool_if:NTF \l__egreg_range_compare_bool      {       % if the digits coincide, print nothing       \str_if_eq:eeF { \str_item:nn { #1 } { ##1 } } { \str_item:nn { #2 } { ##1 } }        {         \str_item:nn { #2 } { ##1 }         % at the first differing digit, don't compare any longer         \bool_set_false:N \l__egreg_range_compare_bool        }      }      { \str_item:nn { #2 } { ##1 } }    }  } \ExplSyntaxOff  \begin{document} \setcounter{page}{95}\index{Topic0|(}Text\newpage \setcounter{page}{101}\index{Topic0|)}Text\newpage \setcounter{page}{124}\index{Topic1|(}\index{Topic2|(}Text\newpage \setcounter{page}{126}\index{Topic1|)}Text\newpage \setcounter{page}{134}\index{Topic2|)}Text\newpage \setcounter{page}{141}\index{Topic0}Text\newpage \index{Topic0}Text\newpage \index{Topic0}Text \printindex \end{document}   yields the following error message:  ! LaTeX3 Error: Too many '--' tokens when trying to split argument.  For immediate help type H &lt;return&gt;.  ...                                                l.3   \item Topic0, \checkrange{95--101, 141--143}   Entering non-stop yields the following output:    which omits "-3" from the second index entry of Topic0. The question is how to get around this.  Edit following up on egreg's ingenious solution.  I don't know if I should ask a new follow up question, so I'm posing it first here in this edit.  egreg's solution doesn't seem to handle index sub-topics.  If one adds the following 4 lines  \setcounter{page}{135}\index{Topic0!sub0|(}Text\newpage \setcounter{page}{136}\index{Topic0!sub0!subsub0|(}Text\newpage \setcounter{page}{139}\index{Topic0!sub0!subsub0|)}Text\newpage \setcounter{page}{140}\index{Topic0!sub0|)}Text\newpage   after  \setcounter{page}{134}\index{Topic2|)}Text\newpage   in the solution, one gets the following errors:  Output written in mwe.ind. Transcript written in mwe.ilg.  (mwe.ind ! Extra }, or forgotten \endgroup. l.4     \subitem sub0, 135--140}  ? r OK, entering \nonstopmode... ! Extra }, or forgotten \endgroup. l.5       \subsubitem subsub0, 136--139}  ) [145] (mwe.aux) )   and this output:    Alas I don't understand egreg's code well enough to make the required changes. Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$ What makes a photo a good photo? I don't know if this is a stupid question, this just came across my mind.  Imagine a photo of a dog yawning.  Someone could say that is a very good photo as it is taken just at the right moment, from a perfect angle (eg. level with the dog) and it has a shallow depth of field focusing on its face.  Someone else might dislike it because unfortunately the dog is ugly.  So from that example I could say people look at the following elements when judging a photo (but I could be wrong):   The timing (if it's taken at the right moment) The angle it's taken from (composition?) DOF (composition?) The subject (eg. whether the dog is pretty)   I am interested to know what experts think the elements you look at when judging whether a photo is a good photo.  Thanks in advance for your comments. I've seen this message several times whenever someone has a problem installing, upgrading or removing some piece of software, but I wonder, what does it mean, and more importantly, is possible to solve it?  (Reading database ... 81657 files and directories currently installed.)  Removing mongodb-10gen ...  arg: remove  invoke-rc.d: unknown initscript, /etc/init.d/mongodb not found.  dpkg: error processing mongodb-10gen (--remove):   subprocess installed pre-removal script returned error exit status 100  invoke-rc.d: unknown initscript, /etc/init.d/mongodb not found.  dpkg: error while cleaning up:   subprocess installed post-installation script returned error exit status 100  Errors were encountered while processing:   mongodb-10gen  E: Sub-process /usr/bin/dpkg returned an error code (1)   (the above is just an example, not my actual problem) using nginx with SNI By now I've not used SNI with nginx yet. But as IP address pools are quite filled and commercial XP support is about to cease (finally) I'm thinking about converting a few sites to SNI.  I'm aware of the general limitations and pitfalls that might come along with SNI (XP issue, very old browsers). But beyond that is there anything I should be aware of?   Like - nginx related pitfalls when using SNI - issues/bugs with recent (notable!) browsers Minecraft Vanilla 1.9 crash with problematic frame ig8icd64.dll Already I have lost all hope of looking at late answers etc. It's just a parade of meh. I can't make any kind of intelligent decision because I can't see the other answers to the question, for instance. There's no Looks Good, only Skip, and those feel different to me. So I've been more interested in .  But people keep approving crap while I'm looking at it. Two or three "the post was already approved; please visit to edit" and I'm just "screw this whole thing".  Can we do something like one reject bounces all existing votes: you need three approves in a row? Because a lot of people are stupidly approving and actually driving me away from the queue.    Note this is less about letting bad edits through and more about keeping reviewers motivated. To make a JavaScript class with a public method I'd do something like:  function Restaurant() {}  Restaurant.prototype.buy_food = function(){    // something here }  Restaurant.prototype.use_restroom = function(){    // something here }   That way users of my class can:  var restaurant = new Restaurant(); restaurant.buy_food(); restaurant.use_restroom();   How do I create a private method that can be called by the buy_food and use_restroom methods but not externally by users of the class?  In other words, I want my method implementation to be able to do:  Restaurant.prototype.use_restroom = function() {    this.private_stuff(); }   But this shouldn't work:  var r = new Restaurant(); r.private_stuff();   How do I define private_stuff as a private method so both of these hold true?  I've read  a few times but it doesn't seem like "private" methods can be called by public methods and "privileged" methods can be called externally. When will data.SE get updated data? What is the name for the grammatical device of putting "not" after a verb to negate it? How to play MP3 files using Rhythmbox? Can I travel in UK before switching to a study visa? As I read in general visitor visa Terms &amp; Condition, I would only be granted to stay up to 6 months in UK. However I am planning to apply for General Student Visa to further my studies. Would it be possible for me to enter UK as visitor for first 6 months/less before I apply for student visa in the same year if I want to travel and study on the same year? In attempting to wrap my mind around the basic vocabulary, concepts, and methods of philosophy, I find myself wondering what the difference is between a philosopher and a theologian.   can have two meanings: 1. Theology is a rational study of the existence of God/gods and the nature of religious ideas. 2. Theology is simply a study of a particular religion (or all religions), really more the practice than the theory, but maybe a mixture.  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It's my understanding that the phrase "Winter is Coming" is the motto of the Stark house, when the story first reaches the wall it's implied that the winter is not a periodic season like it is in our world.  Does the winter refer to a particularly severe season or does a winter only occur every few years? I'd like to have a horizontal line extending from the end of the text on a line to the right margin. So far I've found \hrulefill and \leaders\hrule\hfill, but these draw a line that's flush with the baseline of my text — how can I get it to be vertically centered? Why is "that"/"which" omitted in this sentence? I want to add additional items to the data returned as . So for every function/query that returns a WP_Post object I want my additional data added.  Example of returned result:  WP_Post (object) =&gt; [     // Default data returned by WP_Post     ID                  =&gt; int,     post_author         =&gt; string,     post_name           =&gt; string,     post_type           =&gt; string,     post_title          =&gt; string,     post_date           =&gt; string,     post_date_gmt       =&gt; string,     post_content        =&gt; string,     post_excerpt        =&gt; string,     post_status         =&gt; string,     comment_status      =&gt; string,     ping_status         =&gt; string,     post_password       =&gt; string,     post_parent         =&gt; int,     post_modified       =&gt; string,     post_modified_gmt   =&gt; string,     comment_count       =&gt; string,     menu_order          =&gt; string,      // Additional data I want to add     extra_data_1        =&gt; array,     more_data_key       =&gt; string,     another_added       =&gt; string ]   For example when the functions  or  are run they will return the WP_Post object along with my additional data.  I've tried finding the appropriate  but this has been unsuccessful.  I am hoping I can do something like:  // This is concept code add_action('pre_wp_post_return', function($data) {     $data-&gt;extra_data_1     = get_post_meta($data-&gt;ID, 'extra_data');     $data-&gt;more_data_key    = get_post_meta($data-&gt;ID, 'more_data', true);     $data-&gt;another_added    = get_post_meta($data-&gt;ID, 'another_data', true);      return $data; });   My reasoning is I am having to build a custom API for WP that uses a range of different core functions which return WP_Post objects. Being able to add my additional data in one place would prevent me from duplicating code.  I hope this is clear enough. Any help would be great! Does anybody know how to add the shapefile name to a field in the file's attribute table?  I've found a description:     'You have to use inline variable in CalculateField like %Name% in   expresssion so that whatever the name of the file is, the string will   be inserted based on Name of the dbase file !'   I've tried to CalculateField tool with %Name%, but I got message:  Executing: CalculateField "2011-11-6 15_41_point" DBFName %Name% PYTHON # Start Time: Thu Dec 08 23:34:23 2011 WARNING 000405: No records within table Succeeded at Thu Dec 08 23:34:24 2011 (Elapsed Time: 1.00 seconds)   Where is my mistake? Adverb between conjugated verb and infinitive? How many maximum CPUs does Ubuntu support by default?