I saw a previous proof on this site that addresses this question, but the proof leaves off with a number of form $4k-1$ can't divide $t^2+1$. Why is this? Thanks.
Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$   I guess, that x cannot be even. because we can apply mod 4 test  say $x=2k$ then $y^2=8k^3+23\equiv3\mod(4)$   but, this is not possible, since the square of an integer is congruent to $0$ or $1$ $\mod (4)$  so $x$ is of the form $4k+3$ or $4k+1$  $x=4k+3$ is also not possible for the same reason but for the last case the test fails.  any hints ?
How do I create an .exe for a Java program? Shell script works when saved with nano but not when saved with Notepad++ When I copy my bash script across from Notepad++ into a new file with nano editor inside SSH and save. It runs fine. (sh ./install).   But if I save the file (Exact same contents), upload to my web server, download it using Wget on the same machine. I get syntax errors. I've checked the encoding and they seem to be the same. I have since used a plethora of character encoding to see if it will resolve the issue. I am also setting the file to executable once I download it using wget!  The file runs fine and I have zero errors when copy and pasting using nano. Any idea what this may be? Big O, how do you calculate/approximate it? Most people with a degree in CS will certainly know what . It helps us to measure how well an algorithm scales.   But I'm curious, how do you calculate or approximate the complexity of your algorithms? Is it possible to open directories as administrator? I want to have a root privileges on some directories. How can I do this? Is there a standard place for placing custom Linux scripts? Singular or plural verb after parenthetical material containing a conjunction? I would tend to say     Sara (and all of us) have a lot to think about now.   ...but it occurs to me that if I remove the parenthetical it's clearly incorrect. Is it proper to say     Sara (and all of us) has a lot to think about now.   ...that sounds really strange to my ear.  Also, does it change the rules in any way if I use an em dash, e.g.     Sara–and all of us–have a lot to think about now. Some time ago, we announced that . This has gone well and we're extremely thankful to everyone for their support and feedback. Because it has been successful so far, we're going to be extending the program to the majority of our graduated sites. Well, we've pretty much already done that, but there are quite a few sites that have ads enabled, yet still display nothing. The thing is, we don't really have an abundance of clients offering products that a  would find relevant, but Amazon has stand mixers, a selection of super fine chinois, deliciously sharp cutlery and enough cast iron to fill the Gulf Of Mexico with . Beginning as early as this week (week of 5 February 2018), our ad team is going to be hand selecting affiliate ads to be displayed on select sites. The use of inventory will of course depend on the availability of relevant ads to display which means:  Some sites might see very few, if any ads. Some topics simply don't come remotely close to orbiting anything worth advertising. Furthermore, while graduated, some sites simply don't have enough activity for ads to make sense.  Since these are hand-selected, it could take a while for ads to appear initially.  These would of course yield to inventory from clients if clients had products that were relevant to any given topic.   Nothing changes when it comes to the . If you're used to occasionally seeing a community promotion ad, you'll just notice that you'll also occasionally see an ad for something that is likely to be of interest to someone interested in the topic that your site covers. In fact, to be clear, this is simply a continuation of the announcement we made in November of 2016. We just wanted folks to know that we're going to be continuing the program where there's good potential to show relevant advertisements, even if we don't have clients lined up with a demand for it. Where can I give feedback about a certain ad on [site]? Open a meta post on your per-site meta and let us know. Please remember, the bulk of advertisements are an effort to get something small back from folks that hit the sites through search engines and are extremely unlikely to ever become contributors. Their soufflé stopped falling, and we'll probably never see them again. Maybe we can get a few bucks once they realized that copper ions help egg whites stiffen properly and buy a bowl or a proper bloom whisk. However, you're the experts on these topics and we're very likely going to listen to you. What sites, specifically? Are any sites exempt? If you're a former Stack Exchange 1.0 site and / or subsequently joined the network under a special charter, nothing changes - don't panic :) For the rest, we're just extending what we're currently doing with ads on graduated sites to also include those where there's an opportunity to make sensible use of affiliate ads. If a site has less than 15k (average) daily visits, or fewer than roughly 50k combined total posts - it might not yet be time to invest in selecting ads to be shown there. But, part of this exercise is also seeing what works best. I don't really have feedback, just some suggestions Ads aren't just an opportunity to sell stuff, they're an opportunity to really show people that we understand a given topic. That, of course, makes suggestions from people that do that stuff every day quite valuable. Post on your per-site meta and our community / ad critters will take a look. Questions? Comments? Kvetches? Let us know in an answer or comment below. If anything about the plan changes, we'll update this post to let folks know. Etiquette on sending a thank you e-mail to respondents who gave me helpful information Booting from USB and not installing I created a live usb to boot Kubuntu, which has worked well enough. However, when the installation gets to the point of telling me to restart the computer (and it restarts) the installation window pops up again, and the computer is still signed in as a live user. None of the installation or settings appear to have saved.  Any ideas? A slick proof that a field which is finitely generated as a ring is finite It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,\dots,x_n]\to k$ for some $n\in \mathbb{N}$, then $k$ must be finite. Since finite generation as a ring implies finite generation over the prime field ($\mathbb Q$ or $\mathbb F_p$), by Noether normalization it follows that $k$ must be a finite extension of its prime field. In positive characteristic this finishes the job and in zero characteristic, should lead quickly to a contradiction, though I don't see immediately how. Is there an elementary/slick proof of this fact? List all directories that do NOT have a file with a given file name inside How would I go into listing all directories that do not have a file with a given file name inside? e.g. given this tree  /   /a      README      file001      file002   /b      README      file001   /c      file003   I want to list the directories that do not have a file named README, in this case it would be directory /c. How would I do that? I can't think of any syntax using e.g. find. IBus methods do not show up in text entry settings I installed both Korean and Japanese using the Language Support window, and expected Korean (Hangul) and Japanese (Anthy) to show up in the input sources for Text Entry Settings (having both packages installed). However, neither shows up. Apache on MacOS Sierra: AH00557: httpd: apr_sockaddr_info_get() failed for Macbook-Pro.local AH00558: httpd: Could not reliably determine… I updated MacOS Sierra's Apache and PHP, and now whenever I $ sudo apachectl restart I get this:  AH00557: httpd: apr_sockaddr_info_get() failed for My-MacBook-Pro.local AH00558: httpd: Could not reliably determine the server's fully qualified domain name, using 127.0.0.1. Set the 'ServerName' directive globally to suppress this message   Things I've tried so far that didn't work:   Setting ServerName localhost in /etc/httpd.conf Setting ServerName localhost:80 in /etc/httpd.conf Setting ServerName 127.0.0.1 in /etc/httpd.conf Setting ServerName 127.0.0.1:80 in /etc/httpd.conf Ensuring 127.0.0.1       localhost is set in /etc/hosts Repeatedly restarting Apache Electrodynamics textbooks that emphasize applications Please recommend undergraduate-level textbooks on electrodynamics which emphasize practical applications and real life examples. Please describe the book's level and contents and its intended audience in as much detail as possible.   Please provide both applications from everyday life for curious students as well as engineering applications from a physics perspective. Is it possible to "upgrade" from a 32bit to a 64bit installation? As per title, I wish to know if I can, after a successful 32bit setup and consequent software installations, update to a 64bit version.  I know how to partition (actually one of the solutions is to set /, /etc, /home, /var/www, and /opt as separate partitions) and I know that a clean install is way better than a dirty one, yet I would like to know if/how it's possible to do that. Why {are you / you are} making noise? Design problem - Implement an approval process on case, which sends email to approver showing some additional info, which should not be visible to case owner/agent (who clicked on "Submit for approval" button).  Possible solution - All that additional info will not be stored on Case (otherwise agent will also be able to see it). I am creating a child object to case, where all that additional info be stored. We can make sure agent does not have access to that child object (lets call it ChildObj).   In the email template used, we can include info about that child object. From what i understand, standard email templates dont allow cross object reference. We probably need to use visualforce email templates.  Questions -  1) Can anyone see/point out any issues with that design? 2) I am very new to salesforce. Can anyone give me any example of how i will access child record's field? I mean there won't be any lookup or anything to child object, right? How do i access a field on the child object? Any clue would be helpful Transpose Ranges repeated by row I would like to convert several range of columns into rows, and keeping the association to the original col/row source  Sample source sheet  --------------------------------------------------- | Data 1 | 1/1/2018 | 1/2/2018 | 1/3/2018 | etc...365 days | --------------------------------------------------- | name 1  |    25    |   30     |    5     | etc...|  --------------------------------------------------- | name 2  |    2     |          |    4     | etc...| --------------------------------------------------- | etc ... 100 rows   Expected Results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- | name 1  |  1/2/2018   |   30     |    --------------------------------------------------- | name 1  |  1/3/2018   |   5      |    --------------------------------------------------- | name 1  |  etc.. 365 days --------------------------------------------------- | name 2  |  1/1/2018   |   2      |    --------------------------------------------------- | name 2  |  1/2/2018   |          |    --------------------------------------------------- | name 2  |  1/3/2018   |   4      |    --------------------------------------------------- | name 2  |  etc.. 365 days --------------------------------------------------- | etc ... for each change of name pull all the col dates/values   Current results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- |         |  1/2/2018   |   30     |    --------------------------------------------------- |         |  1/3/2018   |   5      |    --------------------------------------------------- |         |  etc.. 365 days                      I have to copy down to bring in "name 1" for each date  --------------------------------------------------- | name 2  |  1/1/2018   |   2      |             now update the formula to bring in the second item for the sequence of dates --------------------------------------------------- |         |  1/2/2018   |          |    --------------------------------------------------- |         |  1/3/2018   |   4      |    --------------------------------------------------- |         |  etc.. 365 days --------------------------------------------------- | etc ...     =Sheet1!A:A I can bring in the name from the source list, but I have to copy it down for the number of dates.  How can I have it match the number of dates?  and be dynamic to update as dates/values are added Edit: trying to use split and rept to count the number of times to repeat, then transpose.. =transpose(rept(split(arrayformula(concatenate(Sheet1!A:A&amp;"1")),"1",True, True),COUNTA(Sheet1!I1:ZZ1)))  =TRANSPOSE(Sheet1!$C$1:$Z$1) I can use transpose to get the dates/values, but it's fixed range.  How can I make it dynamic to import dates as they are added? General question I have an app X installed on my device in version Y. I don't want to update it to any newer version (Y+z) for some reason. Still, the Google Play Store app always lists updates for X, which keeps me from a simple &quot;update all&quot;. My question, put to a single sentence:  How to hide updates for X permanently from the &quot;available updates&quot; list?   Specific examples First, I know there are alternatives as e.g. the , which can handle this quite well (skip this update, skip all future updates). I prefer a &quot;generic solution&quot;, which does not depend on a 3rd party's market app. Second, I know that  includes a &quot;Market doctor&quot;, which can be used to disconnect an app from Google Play.1 However, in my specific case that only worked for a few hours: I disconnected GTalk, as I don't want it replaced by Hangouts. The update notification disappeared for a few hours, then it was back. No idea what restored the &quot;link&quot;. Third, just to make sure: I'm not talking about the &quot;auto update&quot; feature as discussed e.g. in . I was rather hoping for something close to , but without the need to &quot;freeze&quot; the app in question (which I still want to use, but in the version installed, not in a &quot;future version&quot; available). Fourth: No, uninstalling the app in question is no option here, though it would perfectly hide any future updates from being displayed :) Nor is &quot;freezing&quot;, as the app is used regularly. Fifth: There is a related question for my GTalk example:  However, I want the opposite: No update.  It might very well be that &quot;Google Apps&quot; (and other pre-installed apps) are handled in some specific way, and restore their &quot;market links&quot; themselves somehow. For this case, my question includes: How to prevent that from happening?  Update There's a  discussing the very same issue. Until recently, a work-around for &quot;user installed apps&quot; was to uninstall them and then sideload the .apk – but even that stopped working. Well, re-signing the .apk with a different key might work for user-apps. But both would fail for pre-installed apps (can't sideload them to override; different signature would refuse that install completely). Moreover, there's an  on this by some &quot;joel.bou..&quot;. Not hard to guess, though blanked out: that would be Joël Bourquard. If it's really him, then that's really an issue (for those who don't know: Joël is the dev behind Titanium Backup). Let me quote from the issue:  For the record: in the Android Market days it was possible to &quot;Attach&quot; or &quot;Detach&quot; apps (on rooted devices) with a system tool such as Titanium Backup. But nowadays (with the Google Play Store app) this has become nearly impossible to do, because the complete app list seems to sync from Google's servers and the user has no control over that.  Which explains why detaching with apps like Titanium Backup is no longer permanent. And I'm afraid I've asked the impossible. But I thought the same in the past with other questions, and surprisingly got solutions here. So I do not yet give up hope on this one!  1: what that does is explained by eldarerathis  (thanks to  for digging that up!) Finding a replacement for the scoreboard player test