Prove $a^n+1$ is divisible by $a + 1$ if $n$ is odd:  We know $a$ cannot be $-1$ and the $n \in \mathbb{N}$. Since $n$ must be odd, we can rewrite $n$ as $2k+1$. Now we assume it holds for prove that it holds for the next term.  $$a^{2(k+1)+1}+1$$ $$=a^{2k+3}+1$$ $$=a^3\cdot a^{2k}+1$$ $$=(a^3+1)\cdot a^{2k} -a^{2k}+1$$  Im not sure on what to do next. Since $a^{2k}$ means that the exponential term will be even and thus you cant use the fact that $a^n+1$ is divisible by $a + 1$ if $n$ is odd.
I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. But I found that: \begin{align*} (a-b)(a^{n-1}+a^{n-2}b + a^{n-3}b^2+...+b^{n-1}) &amp;= a^n-b^n. \end{align*}  Doesn't this imply that $(a-b) \mid (a^n-b^n)$ as $a^{n-1}+a^{n-2}b+\dots+b^{n-1}$ is clearly an integer? This obviously isn't a proof by induction, but is there anything wrong with taking this approach to prove this result, other than the fact that it isn't what is being asked?   Aside, I'm having trouble getting started with the proof by induction. I've tried making the above equivalence useful during the induction step, but it doesn't seem to be helpful, so any hints would be greatly appreciated!
download file on form submission I have created a form which has only a submit button. In the submit function I am generating csv file and saving it to the files folder using fputcsv function.  But Now I need extra functionality of submit. After saving the file it should download the file also.  How to do this? Any one have some idea about this? What can I use to find duplicate photos, including photos that have been resized? Why do only Hobbits use a combination of first- and last names? While all the Hobbits I can think of have a combination of first- and lastname (e.g., Bilbo Baggins, Peregrin Took, the Sackville-Baggins's,...), I can't think of any character* of another race that has a lastname:   Men seem to like to add a reference to their ancestors (e.g., Aragorn, Arathorn's son or Isildur's heir) Dwarves seem to like descriptive or honorary names (e.g., Thorin Oakenshield, Dain Ironfoot). They are not lastnames though, since they do not get passed on to descendants. Elves don't have a need for lastnames, since elvish names are unique. Some nevertheless seem to have an additive (e.g., Legolas Greenleaf). Likewise, I can't find any usage of lastnames among Ents, Eagles, Istari, Orcs, trolls,...   But why would a society that is quite rural and not a leader of modernization be the only one to use a modern naming system with first- and lastnames?  *Except for Tom Bombandil. But he is, well, "special". $n!+1$ being a perfect square I'll like to add this functionality to the autocomplete widget in Entityreference field to show node ID next to the titles in the dropdown list. The reason behind the idea is to differentiate between multiple nodes with same title.  Example:    This is a title (3) This is a title (2) This is a title (1)   I know the node ID is shown once a selection is made but I like to show it in the dropdown list in order to pick the right node quickly based on the node ID. Do the druid's uses of the Wild Shape feature recharge while still in Wild Shape?  Say a 10th-level Circle of the Moon Druid spends two uses of Wild Shape to transform into an Air Elemental, whose form she can stay in for the next 5 hours. After 2 hours, she takes a short rest, but does not drop the Wild Shape. She has another couple of hours to keep the Air Elemental form.  My main concern: Does she get her two uses of Wild Shape back with that short rest? Or, does she have to drop the Air Elemental form first before taking the short rest to recharge Wild Shape? "Looking forward to hearing from you" - May I omit "I am" at the beginning? At the end of a letter, is it correct to omit "I am" in this sentence?     I am looking forward to hearing from you   Then it will become:     Looking forward to hearing from you $\operatorname{Ext}$ and injectives, respectively projectives I have gcc 7.3.0 on Ubuntu 18.04. But I am working on a project where I need gcc less than version 6. When I tried to install gcc 5.4 using sudo apt install gcc-5.4, I got this error:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package gcc-5.4 E: Couldn't find any package by glob 'gcc-5.4' E: Couldn't find any package by regex 'gcc-5.4'     What is the appropriate way to install gcc 5.4? I realize this is probably going to come off as a bit silly, but in Iron Man II how could James Rhodes easily don the Mark II armor that was designed to fit Tony Stark?  As you can see in this image (which isn't an "in universe" shot) Tony and Rhodey are roughly the same height, but it appears that Rhodey has a smaller frame than Tony, which would seem to make for an ill fit of the armor, especially for flight.  This issue could of course have been addressed by the Air Force during the retrofit with Justin Hammer. What gifts would be welcome for a Swedish host family? unable to change php version on cli How to highlight (color/draw attention to) a particular snippet in Minted I'm currently doing some work on error messages in programming languages. As such, I'm showing faulty pieces of code, alongside the compiler-errors accompanying them.  I'm using minted for the code snippets, but ideally, in addition to normal syntax highlighting, I'd like to highlight the sections of the code at which the error is located, by underlining them, putting a red box around them, shading their background, etc.  Is this possible using Minted? mysql doesn't ask for root password when installing using trial and error in math problems Suppose one bacterium is in a jar at 12AM, and then suppose there are two bacteria at 12:30AM,  and then there are four bacteria at 1AM, etc.  (a)  How many bacteria will be in the jar at 12PM that day?  (b)  Use trial and error to estimate at what time there will be 1 billion (1,000,000,000) bacteria in the jar.  for a) is it going to be 44 bacterias ? or am i wrong? Please help with b), i have no idea how to figure it out in short term Thanks I'm using a curve modifier and an array modifier. My object is at one end of the curve but as soon as I choose the curve for the curve modifier it moves to the center. How can I fix the problem so that I can have the array going from one end of the curve to the other? Will my checked luggage freeze in the airplane? I'm taking some freeze-sensitive items in a checked luggage for a short flight. It's cheese, chocolate, toothpaste and wine. I would like to know whether the luggage can get "frozen" in the aircraft. Sometimes it happened to me that my stuff actually was quite cold after the flight, so I'd like to be quite sure. Especially the cheese was really expensive and freezing it would very likely demage it.  Usually I pack everything well in clothes, but this time the things are just too many so it's tough to do more than just protect the wine from breaking.  Some details: It's an A320 flight taking 1.5 hours, with Czech Airlines, from Paris CDG to Prague PRG. How to print only last column? echo -e 'one two three\nfour five six\nseven eight nine' one two three four five six seven eight nine   how can I do some "MAGIC" do get this output?:  three six nine   UPDATE: I don't need it in this specific way, I need a general solution so that no matter how many columns are in a row, e.g.: awk always displays the last column. Complete classification of the groups for which converse of Lagrange's Theorem holds Automorphism of rational numbers Let $\mathbb Q$ be group of rational numbers under addition.  Then I want to find out the automorphism group of $\mathbb Q$. In a book it is given that Aut($\mathbb Q$) is isomorphic to $\mathbb Q^*$, the  multiplicative group of rational numbers. I stuck in this.