What are the solutions to the equation: $$\log_{3}\left(2^{x}+1\right) = y$$ where $x,y$ are integers. The first few are $x=1,3$. How would I prove that there are either finite or infinite solutions?
This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.  Edit: Of course, $x = 1, y = 1$ is a solution. I am looking for non-trivial solutions.
My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. Now I want to install Windows 8 for dual boot again, but I have no DVD and do download the ISO one needs a product key. That key is not on the back of the laptop anymore but somewhere on the mainboard.  Is there any way to recover the product key from the mainboard using Ubuntu? Prove that $\liminf x_n = -\limsup (-x_n)$ Why does Kylo Ren's lightsaber look different from normal lightsabers? How to configure an app to run correctly on a machine with a high DPI setting (e.g. 150%)? In eevee, shadows are clipped by camera distance Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital Why won't Smart UV Project automatically unwrap a simple chair? I created a very simple chair mesh and I would like to unwrap it so that I can map a texture. I would like to have Smart UV Project unwrap it automatically, but whether I do it that way or try manually, I am unable to unwrap the model.   Is there something I'm missing?    I tried a few guides, but I'm not sure where I'm going wrong.  My File: How do I challenge an audit failure? I've run through Both SO and Meta searches for an answer, as well as reviewing the , and I cannot find a canonical process to challenge an audit failure.  I can find discussions of the single-choice "I Understand" topic that appear unresolved, in that no solution presently exists.  The particular question in point is .  Despite the number of up-votes, I feel that this quite clearly falls into a common closure category: "Here's my problem; how do I write the program?"  Compare this to   I also failed review on the first link I posted above, .  This looks like a case of code review, rather than SO.  I don't want to get rid of audits; I've learned good things from some of my failures, especially at the start.  However, in the past few weeks, a majority of my failures have been highly questionable, and I've drawn more than one suspension from those that are outright wrong by current standards.  If we have a challenge mechanism, please point me in the right direction.  Otherwise, it's simply not worth the annoyance any more, and I'll drop back to merely comments and answers. Example of $f,g: [0,1]\to[0,1]$ and Riemann-integrable, but $g\circ f$ is not? Prove that if $P$ is a $p$-Sylow subgroup of $G$ and $N \triangleleft\&gt; G$ then:    $ PN/N $ is a $p$-Sylow subgroup of $G/N$ $P \cap N $ is a $p$-Sylow subgroup of $N$ Is there a test for the different styles within maths mode?  What I really want is to be able to save the current style to reinvoke it later on, but a \if along the lines of \ifmmode would work just as well.  In the specific example, I want to measure how big something will be when displayed in maths mode, so I do:  \settoheight{\mathheight}{some bit of maths}   except that it complains because the some bit of maths gets put in a box and measured, and the box doesn't know that it's maths.  So instead I put:  \settoheight{\mathheight}{\(some bit of maths\)}   except that then I get the wrong answer if this is used in displayed maths.  So ideally, I want:  \ifdisplaystyle  \let\currentstyle\displaystyle \else  \iftextstyle   \let\currentstyle\textstyle  ...  \fi \fi \settoheight{\mathheight}{\(\currentstyle some bit of maths\)}   and then I'll be happy.  Is there such a test?    Okay, so \mathchoice is clearly the way to go (thanks Martin!), but its behaviour is a little odd and since I want to use it to define a macro depending on which mode I'm in, this oddity is quite important.  Here's the code:  \newcommand{\mathtest}{% \gdef\name{a}% \mathchoice{% \name \gdef\name{b}% }{% \name \gdef\name{c}% }{% \name \gdef\name{d}% }{% \name \gdef\name{e}% }% \name}  \[ \mathtest \]   If \mathchoice behaved like \if, this ought to print ab.  But it prints ae.  All the options to \mathchoice get processed (which makes me wonder a bit how it works, but that's by-the-by).  So, is there an obvious way to make \mathchoice behave a little more like \if? How to compare two TeX codes and mark differences with color I would like to find easy tool, which compares two TeX codes and generates a PDF file of the first code, in which the differences from the second code are marked by a given color. I use winedt 7.0 under windows 7.  I just noticed that latexdiff provides what I need. As far as I understand it is included in Miktex. Is there an interface (menu commands) in winedt 7.0, which allow to work with latexdiff ? Execute vs Read bit. How do directory permissions in Linux work? I'm using a class I've derived from DocumentPaginator (see below) to print simple (text only) reports from a WPF application. I've got it so that everything prints correctly, But how do I get it to do a print preview before printing? I have a feeling I need to use a DocumentViewer but I can't figure out how.  Here's my Paginator Class:  public class RowPaginator : DocumentPaginator {     private int rows;     private Size pageSize;     private int rowsPerPage;      public RowPaginator(int rows)     {         this.rows = rows;     }      public override DocumentPage GetPage(int pageNumber)     {         int currentRow = rowsPerPage * pageNumber;         int rowsToPrint = Math.Min(rowsPerPage, rows - (rowsPerPage * pageNumber - 1));         var page = new PageElementRenderer(pageNumber + 1, PageCount, currentRow, rowsToPrint)                        {                            Width = PageSize.Width,                            Height = PageSize.Height                        };         page.Measure(PageSize);         page.Arrange(new Rect(new Point(0, 0), PageSize));         return new DocumentPage(page);     }      public override bool IsPageCountValid { get { return true; } }      public override int PageCount { get { return (int)Math.Ceiling(this.rows / (double)this.rowsPerPage); } }      public override Size PageSize     {         get { return this.pageSize; }         set         {             this.pageSize = value;             this.rowsPerPage = PageElementRenderer.RowsPerPage(this.pageSize.Height);             if (rowsPerPage &lt;= 0)                 throw new InvalidOperationException("Page can't fit any rows!");         }     }      public override IDocumentPaginatorSource Source { get { return null; } } }   The PageElementRenderer is just a simple UserControl that displays the data (at the moment just a list of rows).  Here's how I use my Row Paginator  PrintDialog dialog = new PrintDialog(); if (dialog.ShowDialog() == true) {     var paginator = new RowPaginator(rowsToPrint) { PageSize = new Size(dialog.PrintableAreaWidth, dialog.PrintableAreaHeight) };      dialog.PrintDocument(paginator, "Rows Document"); }   Sorry for the code dump, but I didn't want to miss something relevant. Is it true that every pair of twin primes >3 is of the form 6n plus or minus 1? Is it possible to have twin primes whose center is not divisible by 6? I need a CD or USB stick, that will install Ubuntu on a completely headless machine. Stick CD in and restart, no keyboard or screen should be involved.  There are a few obstacles:   The language selection menu when you first boot the CD. The fact that the CD menu waits. The installer's asking questions during installation. The Crossbow Expert feat gives three benefits, the first of which is explicitly tied to crossbows and the last of which to only hand crossbows. But in the middle, there is:     Being within 5 feet of a hostile creature doesn't impose disadvantage on your ranged attack rolls   Notably missing is a qualifier like "when attacking with a crossbow". Does this mean that the benefit applies to other ranged weapons and to spells which require ranged attack rolls? Or does the name of the feat itself limit this to  crossbows? Difficulty in creating macro '\newcommand' for the beginning and end of frames in Beamer I would like to create a Latex macro for my beamer presentations. I use beamer frequently, and want to make macros for the \begin{frame} and \end{frame} to require fewer keystrokes. At the beginning of the document I have:  \newcommand{\bfrm}{\begin{frame}}%start new frame \newcommand{\efrm}{\end{frame}}%end new frame   Which compiles fine. I then test the first of the two macros for the beginning of the frame:  \bfrm{New frame testing}     blah blah \end{frame}   Which produces the result I am looking for. I then test the second of the macros:  \bfrm{New frame testing}     blah blah \efrm   Fails to compile, saying that there is a runaway argument. What is the reason for the failure? they were question banned; it turned out it (in part) it was because of deleted questions.   was deleted by the Community user.  I'm aware it will delete old posts that have a negative score, but for a minute or two, I was really puzzled as to why it was unilaterally deleted without some notation left.  Can we have the Community user leave a notation in a post's history as to why the post was deleted?    Something like the following would suffice:     Post deleted automatically due to having negative votes and no activity for three months Add a link to the timeline of a post For the past year I have been quite active on SO and a bit on meta, but I had never heard of the timeline of a post until today in a comment on meta.  There was an  two years ago of this feature as "experimental". ( for that question).  This feature was , answer: this exists, but the link has to be built manually.  I think this could be quite interesting sometimes and I would like to use this, but I will not do so if I have to build the link each time by myself.  This feature has existed for two years and is still "experimental"? Who is using it? Who knows about it?  My feature request: Implement a link in the posts to their timeline. I am sure this nice feature will be used.