Prove that there aren't any positive integers $a,b>2$ such that $a^2 \vert b^3+1$ and $b^2 \vert a^3+1$
${a^2}{\mid}({b^3} + 1)\;$and$\;{b^2}{\mid}({a^3} + 1)$
Are Mersenne prime exponents always odd? I have been researching  so I can write a program that finds them. A Mersenne prime looks like $2^n-1$. When calculating them, I have noticed that the $n$ value always appears to be odd. Is there confirmation or proof of this being true? What do you call somebody who asks a question and somebody who answers a question? What do you call somebody who asks a question and somebody who answers a question?  I have exhausted the thesaurus with no real luck... any ideas?  EDIT: It is in reference to this - or any other StackExchange - site in which the question is posed by X and answer is submitted by Y.    EDIT 2: In the context of StackExchange, and the fact I need plurality (something I neglected to mention oooops), I have decided upon "Posters" and "Respondents"... Let's say that there exists some "true" relationship between $y$ and $x$ such that $y = ax + b + \epsilon$, where $a$ and $b$ are constants and $\epsilon$ is i.i.d normal noise. When I randomly generate data from that R code: x &lt;- 1:100; y &lt;- ax + b + rnorm(length(x)) and then fit a model like y ~ x, I obviously get reasonably good estimates for $a$ and $b$.  If I switch the role of the variables as in (x ~ y), however, and then rewrite the result for $y$ to be a function of $x$, the resulting slope is always steeper (either more negative or more positive) than that estimated by the y ~ x regression. I'm trying to understand exactly why that is and would appreciate it if anyone could give me an intuition as to what's going on there. Trigger for avoid insertion from data loader? i have scenario like write a trigger avoid insertion of records  from  data loader,only manual insertion trigger will fire?on any object     while Entering records from data loader trigger should not fire? Why is a black hole black? In  (ignoring Hawking radiation), why is a  black? Why nothing, not even light, can escape from inside a black hole? To make the question simpler, say, why is a  black hole black? I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus.  I was thinking of trying abstract algebra. I would appreciate any book recommendations.  Thanks in advance.  Andrew I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot! How do I make a program auto-start every time I log in? I frequently need to start several programs that I use every time I start my computer. How can I make it so that whenever I login the program is automatically launched? Egg replacer for bread dough? I've been reading (most) questions and answers (here) about baking bread (which I love) and how to replace eggs in many recipes (which I wish I didn't need). Turns out, I too have egg allergy (not too severe) and a few of my favorite breads require one or more eggs.  So far I've seen two options, in many threads, that look promising: Ener-G egg replacer and replacing eggs with flax. However, not a single use case I've seen here is about bread. I'm looking for an option to replace 1-3 eggs in a bread dough with: 500 g of white wheat flour, 200 ml of milk, 50-80 g of butter, a variable amount of sugar (5-50 g) and 20-30 g of fresh yeast.  Examples of this dough are the , the  and a similar recipe from Galicia (northwest Spain) with even more eggs. And so many other breads, but these are my favorite ones :)  Has anyone tried flax or ener-g on a similar bread dough? I'm programming a presentation in LaTeX using the beamer class.  However, it defaults to 4:3 aspect ratio slides, while everything I use is 16:9. Is there an easy way to change this using a command or two? If $S_n$ has cyclic subgroups of these orders or not? This question was asked in my quiz and I am having difficulties in solving it .  Question : For a positive integer $n$ , let $S_n$ denote permutation group on $n$ symbols. Choose correct statements from below. (a) For a positive integer $n$ and for every $m$ with $1\leq m \leq n , S_n $ has a cyclic subgroup of order $m$; (b) For every positive integer $n$ and for every $m$ with $n&lt;m &lt;n!$, $S_n$ has a cyclic subgroup of order $m$; (c) There exists positive integers $n$ and $m$ with $n&lt;m&lt;n!$ such that $S_n$ has a cyclic subgroup of order $m$; (d) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of order $S_n$ .  Attempt: (a) is true as I can choose an element of length $m$ which will generate cyclic subgroup of order $m$. (b) Choose $n=4$ and $m=19$ to contradict (b). (c) I dont know if there can be found any such case and if yes how to do so . (d) I dont know about it . kindly tell. Answer:   A,C,D are true How to Async.AwaitTask on plain Task (not Task)? Example of a ring without Invariant Basis Property Determine the Galois group of the polynomial $(x^3-2)(x^3-3)(x^2-2)$ over $\mathbb{Q}(\sqrt {-3})$ Determine the Galois group of the polynomial $(x^3-2)(x^3-3)(x^2-2)$ over $\mathbb{Q}(\sqrt {-3})$.   I tried to decompose the splitting field into a tower, but then it is very complicated in each step and since the degree is large, I cannot determine the group by guessing. It seems to me that Kummer theory might work, but there are $2$-degree and $3$-degree factors. Time travel short story where dinosaur doesn't taste like chicken I ran across this  and it reminded me of an old short story I read a long time ago.  I read it in a collection of short stories.  It would have been a book because I borrowed it from a school library.   Probably in the 1970s or 1980s.  The book and the story would have been older, though.  Probably from the 1960s or earlier.  In the story, a scientist has a machine that he can use to view the past.  It isn't like a TV or something, though.  It opens a portal into the past that he can look through.  It is very difficult to control, though.  Pointing it at a place and time is likened to working in a hurricane.  The portal jumps and moves about randomly.  The steadier he can hold it, the clearer the past appears.  When it is really steady, he can hear sounds from the past.  It never gets really steady enough to reach through.  They try pushing tools through the portal to pick up things in the past.  The steadier the portal, the easier it is to push tools through.  It never gets steady enough for them to push a tool all the way through, though.  It just gets easier to make progress.  One day, they have the portal open and it is acting really wild.  At some point the workers take a break.  The portal is still open, and while they are loafing the portal settles down by itself and becomes steadier than it has ever been.  So steady that a piece of the past falls through.  The portal scooped a chunk of the ground out of the past.  The chunk of ground contains a nest with dinosaur eggs.  The scientist  rescues the eggs, and they manage to keep them warm enough until they hatch.  The dinosaurs are some kind of harmless things, and become the lab mascots.  Always wandering the grounds, and in and out of the lab.  One day, the have the portal open again and one of the dinosaurs wanders into the lab and walks into some high voltage terminal.  Crash, bang.  Short circuit.  All the equipment to open the portal is destroyed, and the dinosaur is roasted from the flash.  The roasted dinosaur smells good, like chicken, and the scientist's son (who has been telling the story) takes a bite to see what it tastes like.  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I've written a pretty simple script to test this:  #!/bin/bash echo "Normal Text." echo "Error Text." &gt;&amp;2   I run this script like this:  ./script.sh |&amp; sed 's:^:\t:'   Presumably, only the lines printed to STDERR will be indented. However, it doesn't actually work like this, as I see:      Normal Text.     Error Text.    What am I doing wrong here? In Magic: The Gathering, the rules for a player losing in a multiplayer game are quite complex, so I'd like to try to understand them by example. I would be happy to ask each of these situations as a separate question if that is preferred, but they all seem related, and I would love to have one place to look up all of these answers.  Let's say I'm playing with two friends, Bob and Chloe, in a simple free for all. Unfortunately, I lose. When I lose, suppose the following:   I controlled an  which has exiled Bob's , does he get his bear back? I have a  on the stack targeting Chloe, does Chloe take the 2 damage still? Earlier in the turn I cast  to prevent damage to one of Bob's . If it later takes 3 damage, will that damage be prevented? Something I controlled gave Bob a 1/1 soldier token. What happens to the token? Something Chloe controlled gave me a 1/1 soldier token. What happens to the token? I had a  enchanting one of Bob's army of . What happens to the bear? I had also played  on Bob's final . What happens to the last bear? Chloe had  my . What happens to the Clone? Are there any other situations specifically addressed by the rules? Distorted circles in axis environment with TikZ With a fresh installation of TeX Live (updated just before writing this post), circles in axis environments are distorted. Consider the following MWE:  \documentclass{article} \usepackage{pgfplots} \pgfplotsset{compat=1.16}  \begin{document}  \begin{tikzpicture} \begin{axis}[   ymin=-3, ymax=3,   xmin=-3, xmax=3 ]  \draw (1,0) circle [radius=1.5]; \fill (1,0) circle [radius=1];  \end{axis} \end{tikzpicture}  \end{document}   The output looks like this:   Obviously, the circles aren't even circles. I haven't encountered this problem before. Did I do anything wrong while setting up my new system?  EDIT: It's not an issue of using axis equal or not since the axes are scaled more or less equally already. Instead, the circles appear as rotated ellipses that are strongly stretched out in one dimension.  EDIT 2: Here's the output of \listfiles:  article.cls    2019/12/20 v1.4l Standard LaTeX document class   size10.clo    2019/12/20 v1.4l Standard LaTeX file (size option) pgfplots.sty    2018/03/28 v1.16 Data Visualization (1.16) graphicx.sty    2019/11/30 v1.2a Enhanced LaTeX Graphics (DPC,SPQR)   keyval.sty    2014/10/28 v1.15 key=value parser (DPC) graphics.sty    2019/11/30 v1.4a Standard LaTeX Graphics (DPC,SPQR)     trig.sty    2016/01/03 v1.10 sin cos tan (DPC) graphics.cfg    2016/06/04 v1.11 sample graphics configuration   pdftex.def    2018/01/08 v1.0l Graphics/color driver for pdftex     tikz.sty    2020/01/08 v3.1.5b (3.1.5b)      pgf.sty    2020/01/08 v3.1.5b (3.1.5b)   pgfrcs.sty    2020/01/08 v3.1.5b (3.1.5b) everyshi.sty    2001/05/15 v3.00 EveryShipout Package (MS) pgfrcs.code.tex  pgfcore.sty    2020/01/08 v3.1.5b (3.1.5b)   pgfsys.sty    2020/01/08 v3.1.5b (3.1.5b) pgfsys.code.tex pgfsyssoftpath.code.tex    2020/01/08 v3.1.5b (3.1.5b) pgfsysprotocol.code.tex    2020/01/08 v3.1.5b (3.1.5b)   xcolor.sty    2016/05/11 v2.12 LaTeX color extensions (UK)    color.cfg    2016/01/02 v1.6 sample color configuration pgfcore.code.tex pgfcomp-version-0-65.sty    2020/01/08 v3.1.5b (3.1.5b) pgfcomp-version-1-18.sty    2020/01/08 v3.1.5b (3.1.5b)   pgffor.sty    2020/01/08 v3.1.5b (3.1.5b)  pgfkeys.sty pgfkeys.code.tex  pgfmath.sty     pgfmath.code.tex pgffor.code.tex tikz.code.tex l3backend-pdfmode.def    2020-02-03 L3 backend support: PDF mode supp-pdf.mkii epstopdf-base.sty    2020-01-24 v2.11 Base part for package epstopdf epstopdf-sys.cfg    2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live Is it possible to set keyboard shortcuts on a webpage? Sockets: Discover port availability using Java