We need to prove that $z = \cfrac {5^{125 }- 1}{5^{25} - 1} $ isn't a prime number . This form comes out to be the sum of a geometric progression with the first term $1$ and ratio $5^{25}$, with five terms. So it reduces to the form:  $$z=1 + 5^{25} + 5^{50} + 5^{75} + 5^{100} $$  Now you could present it as a difference of two squares, but that turns out to be a tedious task.  Any help after this? Thanks a lot!
Prove that   $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that.
I'm an on and off user of LaTeX so I'm not familiar with every package out there. Having said that, whenever I use LaTeX and need to accomplish a certain goal (i.e. ), I search around and find some kind of solution that involves a package that I didn't know about. I think that's great, I learn something new, and it goes in my template file.  Every now and then, though, my search comes up with a package that, after much wrangling, I find out that it's obsolete. An example of this is just recently I came across the subfigure package and wanted to try it out. Soon after I found out from here that it has become . I usually keep my TeX up-to-date because I like keeping up with new XeLaTeX features.   My question is, is there a map or timeline of what TeX packages should be avoided because there are better ways of doing things? An idea I have is some kind of list of all packages split by what they accomplish, then cancelling out packages whose features are incorporated into and improved upon by newer packages.  Edit: another example is lscape versus pdflscape. The latter implements rotating pages in the final PDF not just for pdftex but also for other drivers such as xetex. The only way I found out about this was Googling latex lscape rotate xelatex, for which the second link leads to the  (PDF).  Great answers so far. I'll check them out and see which one works best for me.  Edit 2: Turns out that my question is a bit of a dupe in the sense that a great answer came up on an unrelated question. The nag package . Gotta try it! Why does pluralization of compound words typically occur in the middle as opposed to the end of the word? As I understand, correct plural versions of passerby and attorney general become passersby and attorneys general.  But why?  With passerby, the the preposition "by" has been combined with the noun "passer" to become a one word noun "passerby".  So why shouldn't this be passerbys?  If I am an attorney general, and you are an attorney general, why are we not attorney generals?  As an American I have always heard it as attorneys general, however  suggests British English does use "attorney-generals" (as well as an interesting 2nd paragraph that further probes my question).  Are there hard rules to this topic?  Or merely generally accepted practices based on locale? If $G$ has no proper subgroup, then $G$ is cyclic of prime order This is something I'm supposed to be able to prove for an upcoming test, but I can't find anything to help me prove this in my notes or the chapter, which is on cosets and Lagrange's theorem. If all I start with is the group having no proper subsets, then that means any subset is the whole group. By Lagrange's theorem, that means the index is $1$, but that doesn't get me anywhere. Where do I start? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen? Is "po-TAH-to" an acceptable pronunciation for "potato"? Immortalized in the George and Ira Gershwin song "Let's Call the Whole Thing Off" is the nitpicking of pronunciation differences:      You like potato and I like potahto,   You like tomato and I like tomahto,   Potato, potahto, tomato, tomahto!   According to  and other sources, the English do pronounce tomato as tə-MAH-toh rather than tə-MAY-toh. But this doesn't seem to be the case when it comes to potato.     (From the O.E.D.)           –  Pronunciation: /təˈmɑːtəʊ/     –  Pronunciation: /pəˈteɪtəʊ/   Is "potahto" actually an acceptable way to pronounce potato? Or is the song just wrong? Broken URL for data.stackexchange.com help page When using data.stackexchange.com I fail to remember if as [post link] is the correct alias to get the url for the post. Luckily the site has an  link. Unfortunately it is now broken. Can this be fixed? In the Ubuntu dock on Ubuntu 18.04, multiple windows of the same application are grouped and orange dots are shown to tell you how many windows of the application are opened. When you click on the icon, a small screen capture of each of the windows is shown.  However, these screen captures are shown in an order that's based on which window was last active on the desktop. This is something that clashes with the way I use the dock. I'm used to mentally mapping the windows to their dock positions. As such, I will often be trying to get to "the top one" and then I end up needing several clicks before I actually get to the terminal (or some other application) that I was looking for.  Is there some way to make sure that the order of these screen captures does not change whenever a window becomes active? When should I use "in" or "on"? As it is common with people from my country, I have an immense difficulty with prepositions in English, especially with the use of in and on.   When the preposition indicates the position of the object it is a bit easier to decide which one to use, but as it requires thinking from me, I still say things like "I put the bill in the fridge", "my pencil is on my bag", etc, even though if I think about it, I would know how to say it correctly.  Of course, even in those cases where the preposition indicates position the decision is not always clear, as is the case for example with "on the bus" (you are not on top of the bus, you are inside the bus) or "he is in bed" (you are not inside the bed, you are on top of your bed).  Other cases are even less clear, and I believe there is no real rule to decide what should be said; the person should just know it. Two examples are "in a bad mood", "you are on my mind".  So my first question is whether there is really no rule to help me. The second question refers to two of the cases where I still don't know whether to use in or on:   good luck on/in your exam in/on the list   The last one is especially important to me, as I have asked a number of native speakers, and each time I get a different answer. The only consensus seems to be the case where you say "on top of the list". Apart from that, some people will say that I should say "his name is in the list" whereas others think I should say "his name is on the list". And yeah, a couple of people did tell me that I can use either.  Can someone help me? One problem I have found when learning calculus is that there are many different ways to denote the derivative. If $y=f(x)=x^2$, then we could write \begin{align} f'(x)&amp;=2x \\ y'&amp;=2x \\ \frac{df}{dx}(x)&amp;=2x \\ \frac{df(x)}{dx}&amp;=2x \\ \frac{d}{dx}f(x)&amp;=2x \\ \frac{dy}{dx}&amp;=2x \end{align} And this is just Lagrange and Leibniz's notations alone. What I find troubling is that they all seem to be suggesting subtly different things about what the derivative actually is. Is it a function, a limit of a quotient, or both? In the interests of keeping my post brief, I'll focus my attention on $f'(x)=2x$ and $\frac{dy}{dx}=2x$, as these seem to be the most common notations.  $$ f'(x)=2x $$  It does make sense to think of the derivative as the gradient function: $$ f'\colon x\mapsto\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In this case the limit expression is equal to $2x$, and so we can write $$ f' \colon x \mapsto 2x $$ However, this notation seems a little counter-intuitive when we consider what it means to differentiate a function with respect to a variable other than $x$. If I ask what is the derivative of $f(x)$ with respect to $\frac{x}{2}$, does this question make sense? Is it simply $f'(\frac{x}{2})$? Or do we have to express $x^2$ in terms of $\frac{x}{2}$? And how can we can express this derivative using Lagrange's notation?  $$ \frac{dy}{dx}=2x $$  There are many things which are nice about Leibniz's notation, including the fact that it is explicit which variable you are differentiating with respect to. However, in this case, it is unclear whether we are talking about a function, or something else entirely. There are other issues. Some people say they dislike the Leibniz formulation of the chain rule $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$ saying that they find it to be inaccurate. I don't really understand why this is the case. Could someone please elaborate? Not every boolean function is constructed from $\wedge$ (and) and $\vee$ (or) Prove that not every boolean function is equal to a boolean function constructed by only using $\wedge$ and $\vee$.  Here is my solution, can I ask for a feed back on my solution please?   $p∧q$                          $0 0 0 1 $   $p∨q$  $0 1 1 1$  Not every boolean function is the same when using $ ∨ $ and $∧.$  Edited part!  $p\vee q$                  $0111$  $(p\vee q)\wedge p = p$  $0 = 0$  $0 = 0$  $1 = 1$  $1 = 1$  $(p\vee q)\vee p = p\vee q$  $0 = 0$  $1 = 1$  $1 = 1$  $1 = 1$  here is my edited answer , can I ask for more feed back , thanks Problem:  Start page numbers from chapter one and also reflect this in the TOC.  Currently the code for TOC is:  \renewcommand{\contentsname}{Table of contents}  \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \tableofcontents   I would appreciate if I can be pointed in the right direction. So far I have understood that you can use \chapter* in order not to include a chapter in the TOC.  Questions:  1) How would you force Latex to start numbering from chapter 1 and forward?  2) How can you start page numbering from the first chapter? Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x? Does there exist a continuous function such that f(x) is rational for every irrational x, and f(x) is irrational for every rational x? I can think of examples where this would be true, but I'm having a lot of trouble with the continuity. The Wikipedia page () proves that $$f(x) = \begin{cases}  \exp(-1/x),  &amp; \mbox{if }x&gt;0 \\ 0, &amp; \mbox{if }x\le0 \end{cases}$$ is a non-analytic smooth function. But I don't understand why the fact that "the Taylor series of $f$ at the origin converges everywhere [where does "everywhere" mean here?] to $0$" implies the Taylor series converges to $0$ when $x&gt;0$? Can someone explain this in greater detail? I guess there should be a gap in my understanding of power series. Every planar graph has a vertex of degree at most 5. I am trying to prove the following statement, any help!?     Prove that every planar graph has a vertex of degree at most 5. The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be differentiable.  How can I illustate this statement using a counter example? Can I use $ f_{n}=\frac1 n \sin nx $ as a counter example?  Also, is $ C^1[0,1] $ is same as the space given by $ X=\{f\in C^1[0,1]:f(0)=0 \}$. Can I use the same example to show that this is not a Banach space?  More help is appreciated! Thanks! For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$ If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. Compute the Riemann integral  $$\int_0^{\frac\pi2}\frac{\mathrm d \theta}{\sqrt{\sin \theta}}$$  It seems very difficult, I don't know how to go ahead.  Thank you very much for your help! I have an uploaded video and I want to cut sections of audio (strong language) or add beeps on top without reuploading the video. Is there any difference between "color" and "colour"? What is the difference between color and colour?