Is there a way to find the solutions of $3^y=2^x-1$ where $(x,y)$ are integer coordinates (maybe within a certain interval)? Are there an infinite number of integer coordinate pairs for this equation? Is it possible to even determine this? I'm not quite sure the difficulty of this problem, but substituting $x$ or $y$ values into this equation to find integer pairs is time consuming and not very promising.
Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$?
I am a non-EU national, permanent UK Resident. Recently, my work issued me with a multiple Schengen visa with Norway as the first (main) point of entry. Now I want to go on holidays to Greece - do I need to apply to Schengen visa again, as Greece is not Norway and I am not going there for work? Selecting all objects that have been specified in Modifiers Could anyone help in making a script for selecting all objects that are specified in the different modifiers that an object has stacked? For example if my object has a Boolean modifier and an Array modifier, which both refer to an empty or another object (in the Array modifier there is for example an Object offset you can specify). Would it be possible to select all those object with one click? Thanks. What exactly is the point of memoryview in Python Is this a plant or a fungus? It was soft and covered in flies. My guess is fungi.  Location: South Florida. What happens to Bruce Banner when Hulk loses a tooth? In Age Of Ultron we witnessed Hulk lose a tooth.   My question is: Has Bruce Banner lost a tooth as well? Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial The problem:  X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given  PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$  Expected value is $\sum{x_ip(x_i)}$  But this is where I get stuck, I'm really rusty on my statistics and I'm not sure exactly how to structure it in the next step? I think I want to get the form of the following out of the summation  $\sum _{k=0}^{n} \binom{n}{k}p^k(1-p)^{n-k} = (p + 1 - p)^n = 1$  But I'm not sure if it should look like  $\sum \frac{1}{xp(x)+1} $  and if it should where to go from here? I've seen some of the other threads on using strings as x axis labels, but nothing that fits my situation.  I'm trying to create a bar chart from a CSV file. I can do it so long as everything in the data is numeric. However, in my actual data table, the first row is multi-word string descriptions. I need these descriptions to be the x axis labels for each bar.  I've tried symbolic x coords, but I get the "could not parse input as a floating point number" error. I've also tried adding an ID column and using that as the x values, and assigning xtick labels as the string descriptors. I get the same issue that way.  Here's an attempt at an MWE:  \documentclass{article}  \usepackage{lmodern} \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable}  \begin{document}  \pgfplotstabletypeset[columns/category/.style={string type},col sep=comma]{Book3.csv} \vspace{1cm}  \begin{tikzpicture} \begin{axis}[ ybar, xlabel=Xstuff, ylabel=Value, symbolic x coords={cat a,cat b,cat c,cat d,cat e,cat f,cat g}, %xtick labels={cat a,cat b,cat c,cat d,cat e,cat f,cat g},% this is from when i tried the xtick label method xtick=data, nodes near coords, nodes near coords align={vertical}] \addplot table[x=category,y=value]{Book3.csv}; %\addplot table[x=category,y=value]{Book3.csv};&amp; again, from trying the xtick label method  \end{axis}  \end{tikzpicture} \end{document}   Since I suspect this might get answered quickly by you chart gurus, I have a follow-up question:  I have to do a bunch of these charts, and I would like to automate the symbolic x coords process. Is there a way to populate the dictionary of accepted x coords by reading a column of the data? (something like symbolic x coords={columns/category/{Book3.csv}} )?  Book3.csv looks like this (ignore the ID column if using the xtick label method):  ID, category, value, value 2 1,  cat a,    1,     7 2,  cat b,    2,     6 3,  cat c,    3,     5 4,  cat d,    4,     4 5,  cat e,    5,     3 6,  cat f,    6,     2 7,  cat g,    7,     1 I am registered on about 80 SE sites, am an "avid user"(>200 rep) on about 30, and a modest user (103-199 rep) on about 20 more.  Once when I tried to ask a question on Open Data (my "new user" site), the "systemwide" 40 minute rate limit kicked in. So I asked on meta , given that my previous post on the site was days ago. So a mod commented by telling me about the systemwide post rule and asking me if I had just come from another site. I said, yes, History (my best site, 40K rep.)  I can see why I would be rate limited on Open Data. I can understand that I would be rate limited if I had come from say, Open Source, another site where I'm a new user. But did the system "rate limit" me as a new user on one site, based on my activity on History, an "established" site?  Also, about the one day new user rate limit after a question is downvoted: Suppose a question on the previous site receives two upvotes and one downvote (net of one upvote). Is that a different situation than receiving one down downvote and no upvotes? Or does the downvote count in both cases, including the one where there were offsetting upvotes? How about if the downvoted question on the previous site was deleted?  I don't have this problem when going from one "established" (greater than 125 rep) site to another. I understand why a person would be "rate limited" going from one new site to another; SE wants people to ask questions 40 minutes apart on their new sites. In my case, I was blocked when I left an "established" site (my best) to go to a new site, which is almost the same as going to a new site de novo.  Why would this be? Or can't the system tell that my "first" site was an established site? Would I have been similarly rate limited if I went first to the "new" site, and 39 minutes later to the established site? Or does the rate limit apply only when the "destination" site is the new site?  Put another way, there are four permutations of established and new sites: 1) established to established 2) new to new 3) established to new and 4) new to established. The block doesn't apply in the first case, does apply in the second case, and apparently applies in the third. What about the fourth? And why? Because it is not "uniform" across sites in the four cases.   Edit: Unlike the other question, this one is not about the general rule. It is about the "special case" described in the paragraph above. Run cmd.exe in Administrator mode without explorer.exe I've recently had trouble with explorer.exe not working.  I found a possible solution , using .  The problem is that I cannot "right-click Command Prompt, and then click Run as administrator", since explorer.exe does not work for even a second without crashing.  I tried to use the  to open cmd as administrator, but I get an error 1327 because I have a blank password.    My question is, is there any other way to run cmd.exe with admin privileges?  If not, is there any way to run sfc.exe without admin privileges?  If not, does anyone have solution to my original problem of explorer.exe stopping and restarting ad nauseam? How do I call one constructor from another in Java? Run a command before & after ubuntu apt upgrade & unattended-upgrades How to remove a partition on a hard disk? Why does no physical energy-momentum tensor exist for the gravitational field? If I have a double (234.004223), etc., I would like to round this to x significant digits in C#.  So far I can only find ways to round to x decimal places, but this simply removes the precision if there are any 0s in the number.  For example, 0.086 to one decimal place becomes 0.1, but I would like it to stay at 0.08. Prove that for any positive integer $n$,       $$n^{1/n}  &lt; 1 + \sqrt{\frac{2}{n}}.$$   This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.   Hint:      Use the binomial theorem. This question has been asked before, at least twice by people who were trying to cheat on the PROMYS Europe 2020 Application, and once in a  because the OP showed no effort. I haven't been able to solve it, so I'm going to to ask it again.  PROMYS applications have closed, and I hope the effort shown below is enough to keep this question from being closed.     The Mathematical Forest is grown in a two-dimensional plane, where   trees can only grow on points with integer coordinates. To start with,   there are no trees at all. The foresters plant the first tree at   $(0,0)$. Each year, they carry out tree planting according to the   following rule. If there is a tree on the point (𝑚,𝑛) but there are   no trees on the points $(m+1,n)$ and $(m,n+1)$, then they can choose   to remove the tree on $(m,n)$ and plant new trees on the points   $(m,n+1)$ and $(m+1,n)$. For an integer $k\geq1$, the $k$th diagonal   consists of all points $(m,n)$ with $m+n=k−1$. Is it possible for the   foresters to arrange their planting so that eventually there are no   trees on the first $2$ diagonals? What about the first $3$ diagonals?    $4$ diagonals? Can you generalize?   It's easy to get to a position with no trees on the first two diagonals.  (It only takes $4$ plantings.)  I can prove it's impossible to reach a position with no trees on the first $4$ diagonals, and I believe it's impossible to reach a position with no trees on the first $3$ diagonals, but I can't prove it.  For the $k=4$ case, I used a pagoda function, as in a peg solitaire problem.  For $k=1,2,3,\dots$ define the potential of a tree on the $k$th diagonal as $\frac1{2^{k-1}}$, and the potential of the forest as the sum of the potentials of all the trees in it.  When a tree of potential $\frac1{2^{k-1}}$ is removed, it is replaced by $2$ trees of potential $\frac1{2^k}$ so the potential of the forest never changes.  Initially, the potential of the forest is $1$.  Since there are $k$ trees on diagonal $k$, the potential of all points beyond the fourth diagonal is $$\sum_{k=5}^\infty\frac k{2^{k-1}}=\frac34&lt;1,$$ so it is impossible that there are no trees on the first $4$ diagonals.  The potential of the fourth diagonal is $\frac12$, so this argument doesn't show that it's impossible to have no trees on the first $3$ diagonals.  I've done a lot of experimenting, though, and I believe that the statement is true.  (I even wrote a little computer game so I could experiment quickly.)  The best I've been able to do is to get down to one man on the third diagonal.   The black circles are trees eligible for removal, and the gray circles are other trees.  You can see that there is a "traffic jam" in front of the lone tree on diagonal $3$, and there seems to be no way to clear it.  I've been trying to come up with an argument that reflects this traffic jam, but I haven't been at all successful. We know that after $n$ removals there are $n+1$ trees, and I've been trying to prove somehow that they can't all be sufficiently far from the origin to allow all trees to get past diagonal $3$, but I haven't come close.   In the diagram above, if the tree at $(1,1)$ were moved to $(0,2)$, and the tree at $(1,2)$ were moved to $(2,0)$, then it would be possible to remove the tree at $(0,2)$, emptying the second diagonal.  In the fictitious position, we still have one tree on diagonal $2$ and two on diagonal $3$.   So, I think an argument along the lines I was trying cannot just be concerned with the distance from the origin; it must somehow take into account how the trees interfere with one another.  I haven't been able to do this.    I'd be grateful for solutions, hints, or counterexamples.  I wasn't really sure what tags to attach.  Please add any that seem appropriate. why are the best questions the ones which have been closed? I frequently come across a really useful answer on StackOverflow which has been closed because it doesn't match the Q&amp;A format.  Examples include  ,  ,   ,  and there are others.  Where should this information be exchanged if it's not on stackoverflow?  It seems the moderators are often somewhat overzealous...  ---edit  When I say best, I mean they are informative and useful for someone who's looking for an answer to them.  Even if they do become a bit of a discussion, I value other people's opinions on the relative merits of the different JPA frameworks.  I come here looking to choose one and the answer helps me.  So what if it's not a definitive answer?  ---edit  I don't care how popular a question is, nor how interesting it is.  Users are interested in stuff that's useful to them.  Plenty of those were useful (as judged by the upvotes on the answers: 'This answer is useful'), but closed because they didn't have a definitive answer, as far as I can tell.  ---edit  Seems I'm not the first person to think this: there are various similar questions to mine like this one:   In it the user lists lots of other useful questions which have been closed:                    So isn't there a place on stackoverflow for these questions?  A discussion section, maybe? How to pass variable number of arguments to printf/sprintf I want to do:  cat update_via_sed.sh | sed 's/old_name/new_name/' &gt; new_update_via_sed.sh   in my program.  But I want to use variables, e.g.  old_run='old_name_952' new_run='old_name_953'   I have tried using them but the substitution doesn't happen (no error). I have tried:  cat update_via_sed.sh | sed 's/old_run/new_run/' cat update_via_sed.sh | sed 's/$old_run/$new_run/' cat update_via_sed.sh | sed 's/${old_run}/${new_run}/' Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$