I have started it like this If $p&gt;2$, then $p= 2k+1$ , $p^2= 4k^2 +4k+1$, $p^2 -1 = 4k(k+1)$ , But I can't get how to make it divisible by $8$.
Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
Characterizing non-constant entire functions with modulus $1$ on the unit circle Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$ I am having a hard time with the following real analysis qual problem.  Any help would be awesome.  Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p&lt; + \infty$. Let $T_r(f)(t)=f(t−r)$. Show that $\lim_{r \to 0} \|T_rf−f\|_{L_p} =0$, that is $$ \lim_{r\to 0} \left( \int_{\mathbb R} |f(t-r) - f(t)|^p \mathrm d t\right)^{1/p} =0.$$ Who fired the shot and what is the symbolism in firing shot right next to Lucas at the end of the movie? At the end of , during the deer hunt, there’s a shot fired right next to Lucas (deliberately missing the shot).   Who fired the shot? Is it Theo's son? What is the meaning of that scene? Is someone warning Lucas that he made a mistake and definitely, Lucas will be punished in the future? Is there a way to take a mesh, cut it with seams or sharp edges and flatten it, so that the faces stay the same shape and size?  Making paper models is one aspect, where you could use that technique.  No, I'm afraid SZ0Return is not going to do it properly.  And misusing the UV-unwrap is only a last option.    Note, while it's an option to revert to Blender 2.49 which had a script to do this, I'd like a solution that I can use inside the current workspace. How can we communicate short messages long-distance? In our 5e game, we stole a boat. And a horse carriage. And a lot of other stuff for moving things between cities.   Our DM meant for this to be a way to skip road encounters. We decided to fund a shipping company and hire managers (Knights retainers, etc.) to turn a profit. The DM ruled that these would cover lifestyle expenses, provide us with easy smuggling, etc.   However, we want to keep tabs on the treasure we're moving back and forth.   What can we use for communicating with our managers from anywhere in the world? What about from say, across the city?  We don't need to communicate often or for long - a single round will work. All we need is for the manager to say "we need you back" Grammar checking program for LibreOffice Is there a tool for checking grammar in LibreOffice? Getting started with neural networks for forecasting Alternative to Adobe Illustrator Can you recommend a good alternative to Adobe Illustrator?   I am learning vector based graphic design, but I am not ready to make such a large investment as to purchase a copy of Adobe Illustrator. Also, Illustrator might be a little more power than I need at the moment. I am using a Macintosh. Using headset controls like a smart phone I have a generic pair of smartphone earphones that have the button (along with a microphone) and I would like to use the button as a pause/play key in Ubuntu, like I would on an Android phones music player.  Is there a way to enable this headset control in Ubuntu?  If the headset controls are a hardware feature of audio jacks used in Android phones, the name of that hardware (audio jack) would be appreciated. Why time is considered a dimension? Why is time considered to be a dimension? And the other 7 (except the 3 dimensions of space, and the dimension of time) dimensions that string theory suggests, why can't they be realized? Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $ How could we prove that this inequality holds      $$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$   where $n \in \mathbb{N}$, I think we could use the AM-GM inequality for this but not getting how? How to escape quotes in shell? I'm having trouble with escaping characters in bash. I'd like to escape single and double quotes while running a command under a different user. For the purposes of this question let's say I want to echo the following on the screen:  'single quote phrase' "double quote phrase"   How can I escape all the special chars, if I also need to switch to a different user:  sudo su USER -c "echo \"'single quote phrase' \"double quote phrase\"\""   Of course, this doesn't produce the right result. If I was at point $A$ and I wanted to walk directly to point $B$, I would have to walk half way to point $B$, but before that I would have to walk half way to halfway to halfway to point $B$ and half of that again and so on and so fourth. if I halved this distance an infinite amount of times then there would be an infinite amount of actions I would need to perform in order to cross from $A$ to $B$. Therefore, if each action took any quantity of time at all, then it would take me an infinite amount of time to cross from $A$ to $B$, even if $A$ and $B$ were only a few centimetres apart!   Because I am not infinitely old and I can move, there must be a flaw in this logic. Where is it? Problem with \pspicture in a simple plain TeX code Just writing the following very simple program I had some problems. I use to write in plain TeX and in this case I tried to make a picture with PSTricks and first of all sized it with \pspicture. The program is:  \input pstricks \pspicture (0,0)(5,5) \endpspicture \bye   but when I compiled with TeX, I encountered the following error:  Command Line:   tex.exe --src --interaction=errorstopmode "figura.tex" Startup Folder: C:\Users\MT\Documents\TeX  This is TeXk, Version 3.14159265 (TeX Live 2015/W32TeX) (preloaded format=tex)  Source specials enabled. (./figura.tex (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.tex we are running tex and have to define some LaTeX commands ... (c:/texlive/texmf-dist/tex/generic/xkeyval/pst-xkey.tex 2005/11/25 v1.6 PSTricks specialization of xkeyval (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex)) 2014/12/03 v2.7a key=value parser (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvtxhdr.tex 2005/02/22 v1.1 xkeyval TeX header (HA)))) (c:/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex `pst-fp' v0.05, 2010/01/17 (hv)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code .tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.te x) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics .code.tex))) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex) )) `PSTricks' v2.64b  &lt;2015/14/11&gt; (tvz) (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.con)) ! Undefined control sequence. \endpspicture ...z@ \box \pst@hbox \else \fboxsep                                                    =0pt \fbox {\rule [-\pst@s... l.5 \endpspicture  ? x No pages of output. Transcript written on figura.log.  _____________________________________________________________________   TeX Compilation Report (Pages: 0)   Errors: 1   Warnings: 0   Bad Boxes: 0 _____________________________________________________________________   What's the problem? How to prove $( A \times B ) \cap ( C \times D ) = ( A \cap C ) \times ( B \cap D )$ ? Isle vs. Island Some islands are called isle like "Isle of Man", "Isle of Tortuga" and the "British Isles". Other islands are called island, like "Island of Malta" or "Island of Cyprus".  What is the difference between the words? How to know if a land mass confined by water is called an isle or an island? What kind of things can you take from a hotel? Hotels normally provide a big amount of sets (well, depending on the quality, but however, many!) that can help you feel more comfortable there.  So for example you can get towels, slippers, candies, pens, shampoos, shaving kits, sponges, ...  The same in the self-service breakfast: pastries, bread, ...  They are meant to be used during your stay. However, they can be provided in such an amount that there are some extra by the time you leave... and make you feel like taking some back home.  My general rule is to take those things that I used but didn't finish. However, sometimes I may be tempted to take the brand new stuff lying there before I close the door for the last time.  So the question that pops is: what kind of things is it correct to take from hotels and what is it not recommended? Else, what items may differ from country to country? Using proof by contradiction vs proof of the contrapositive How we differentiate between flash memory and SD card memory? Why does the light side of the moon appear not to line up correctly with the evening sun?