Find integers $x$ and $y$ such that   $$x^3+x-y^2=1.$$   My try:  $$x^3+x-y^2=1 \implies x^3+x-1=y^2.$$  Now, when $x^3+x-1$ is a perfect square?
How can be solved the equation $x^3+x-1=y^2$ in positive integers?    I know this equation defines an elliptic curve, but this seems to be a non-elementary way to solve this question. Is there a more elementary solution?   By the way I found three solutions: $(1,1), (2,3)$ and $(13,47)$. Is this related to the law group property of elliptic curves? Thanks for further answers.
Speeding up a NOT IN subquery Good evening,  I'm currently up to try to improve the performance of some of my queries. As far as I know, statements like "IN" or "NOT IN" are even faster with a large amount of values if a subquery on an indexed field without conditions is used.  SELECT * FROM table1 WHERE field1 NOT IN (SELECT index_field FROM table2)   When it comes to use conditions at table2, the query becomes realy slow on a large amount of data.  SELECT * FROM table1 WHERE     field1 NOT IN (         SELECT             index_field         FROM table2         WHERE             user_id = '2'     )   I was thinking about using LEFT Join but when I need to to the filtering for the user_id, it also becomes slow.  How may I solve this problem? Currently, I've no clue about that.  The original query looked sth. like this  SELECT     i.* FROM stream_item si LEFT JOIN user__publisher_item ui     ON ui.user_id = 2     &amp;&amp; ui.item_id = si.item_id INNER JOIN item i     ON i.id = si.item_id WHERE      si.stream_id IN (5,7,8,9,19,24,29,42,43,44)     &amp;&amp; ui.id IS NULL || (         ui.`read` = 0 || ui.saved = 1     ) GROUP BY     si.item_id ORDER BY     si.`found` DESC LIMIT 0,10     id      select_type     table   type        possible_keys                                                                           key                 key_len     ref                 rows        Extra 1       SIMPLE          si      index       PRIMARY,stream_2_item,stream_id_found                                                   stream_2_item       4           \N                  663236      Using temporary; Using filesort 1       SIMPLE          ui      eq_ref      user_id_item_id,user_2_item,user_id_read_saved_hidden,user_id_saved,user_id_hidden      user_id_item_id     8           const,si.item_id    1           Using where 1       SIMPLE          i       eq_ref      PRIMARY                                                                                 PRIMARY             4           si.item_id          1      I have a table which represents the n:m relationship between stream and items. An unique item can be accessible via multiple streams. Based upon this i have a state-table for user dependent states of an item. Currently, the user-state-entry will only be generated, if the item was read or saved the first tim. Otherwise, there is no entry in the user_item table. In the beginning, this query was fine but now, it takes a lot of time. Even if I leave out the "INNER JOIN" it takes 7 seconds on the current tables with around 500k rows each.  The table-structure in the background is as follows:   stream stream_item //n:m relation between stream and items item //unique items stream_group //user-defined groups of multiple streams stream_group_streams //_n:m relation between stream_groups and streams_ user user_item //state-table that holds the user-dependent states of an item - if there is no state, no row exists for an item   I already asked this question at  but added my 2 Edits with further information to late so that I haven't got an answer anymore. Does Reincarnation allows skills to be passed into other classes I'm mainly asking this for Disgaea: Afternoon of Darkness, given I have enough Mana to reincarnate a character into 4 different types of Mages (Green, Blue, Red, Star), a Cleric and then a Warrior, would the Warrior have all the spells learned from the previous classes? I am new to Force.com and trying to learn some concepts.  Currently I am using Single Email Message to send my emails, but I need to change that to mass email, but I don't quite understand how to do that.   I have also created a Visualforce email template, which I would like to use, but it doesn't seem to work.   It would be great if someone could help me out.  //email_disease_map contains pairs of email address and disease ID for (Id id : email_disease_map.keySet())   {     Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();     String[] defaultAddresses = new String[] {'id@test.com'};      List&lt;String&gt; toAddresses = new List &lt;String&gt; (email_disease_map.get(id));      if (!toAddresses.isEmpty())       {         //set addresses for email to be sent         mail.setToAddresses(defaultAddresses);         mail.setCcAddresses(toAddresses);         mail.setTemplateId(Notification);         //set up remaining email         mail.setReplyTo('test@mail.com');          mail.setSenderDisplayName('Ms Test');          Messaging.sendEmail(new Messaging.SingleEmailMessage[] { mail });       }  } Is it "I" or "me" in "Keep Tom and I/me updated"? In this case what is correct?        Keep Tom and I updated.      or         Keep Tom and me updated.      I understand that me is an object pronoun. And therefore I feel the second option is correct.  Especially, because I would say "Keep him updated" and also "Keep me updated". Instead of "Keep he updated" or "Keep I updated". But something about saying "Keep him and me updated" feels wrong.  However I am not sure, because several people have told me that in this case I is the correct option because it is being used in conjunction with a name (Tom) and not a pronoun. Is that correct? Programmatically open new pages on Tabs How to change background-color on text links on hover but not image links I have a CSS rule like this:  a:hover { background-color: #fff; }   But this results in a bad-looking gap at the bottom on image links, and what's even worse, if I have transparent images, the link's background color can be seen through the image.  I have stumbled upon this problem many times before, but I always solved it using the quick-and-dirty approach of assigning a class to image links:  a.imagelink:hover { background-color: transparent; }   Today I was looking for a more elegant solution to this problem when I stumbled upon .  Basically what it suggests is using display: block, and this really solves the problem for non-transparent images. However, it results in another problem: now the link is as wide as the paragraph, although the image is not.  Is there a nice way to solve this problem, or do I have to use the dirty approach again?  Thanks, FileReader.js nothing happens in IE9 Do these leaves signify anything? My friend and his wife visited UK during the year 2015. And they came back on time we want to know one information as they same couple want to visit UK again  For how many years or months can multiple entry visa can be given for a couple who have already visited UK? What is 'deictic time'? [5]     i  If she beats him he’ll claim she cheated.    [non-deictic past]     ii If you eat any more you’ll say you don’t want any tea.[non-deictic present]      The preterite and present tense inflections on cheat and do indicate that Tr is respectively anterior to and simultaneous with To, but here  To is clearly not Td. The time of the (possible) cheating is not anterior to the time of my uttering [5i], but to the time of his (possibly) making a claim of cheating.                       — The Cambridge Grammar of the English Language, p126   I can understand To (time of orientation = utterance time), yet don’t get what the ‘deictic time (Td)’ is. Would you let me know the Td? Cann I use a 19.5V 4.62A laptop charger for 19V 4.74A laptop? Google Analytics: How can I track between http:// and https:// profiles for a domain? Windows 7 + Deep Freeze - I'm stuck in an endless reboot loop I have the following setup:   Windows 7 Ultimate Deep Freeze   I "thawed" my machine last night and performed a Windows Update. The update is having issues (it gets stuck at 32%, fails, and restarts my machine). When it reboots it attempts it again, and again, and again, etc. (Endless loop).  I looked online and found some solutions, but none of them seem to be working:   When I run Safe Mode, Safe Mode w/ Network, or Safe Mode w/ Command Prompt it attempts to revert the Windows Update changes. However, the problem is with Deep Freeze on (and now in "Frozen" mode) the reverted changes don't stay, and I'm back into the loop of death. Oh, and side note: "Safe Mode w/ Command Prompt" does not actually take me to a command prompt window? Perhaps because it is attempting to complete the Windows Update changes first? I have tried to select the option to NOT restart when an windows error occurs, but it still does. I tried the remainder of all the other options in the F8 screen.     The only other option left is to find my Windows 7 Media Disc (I can't find it right now) and use it to repair windows (because for some reason the repair option does not show up in the F8 screen).  Is there a way to disable Deep Freeze from loading? When I selected "Safe Mode w/ Command Prompt" I noticed that it loads the DpFrz.sys file. I know that when I'm in the Windows Boot Manager if I press F10 instead of F8 (while highlighting Windows 7) it takes me to an "Edit Boot Options" screen:     Edit Windows boot options for: Windows 7      Path: \Windows\system32\winload.exe      Partition: 2 Hard Disk: 8e90e329      [ /NOEXECUTE=OPTIN (I CAN EDIT THIS LINE)   ]     Update: I found my Windows 7 Media Disk and it did not help out. The laptop had the "System Restore" as a partition on the HDD. I later received (in the mail) a Windows 7 Upgrade Disc from Sony to upgrade my system from Windows Vista to Windows 7 Ultimate. I placed the disc into the DVD drive and it does not come up as a "bootable" disc. I'm going to try to find an alternative disc to see if I can get into Command Prompt.    Update 2: I got a Windows Repair disc and got into a command prompt window. I got into the registry and disabled Deep Freeze. Also:   I renamed the Pending.xml file to Pending.old I cleared out the Windows Temp directory   I still am stuck in the loop (though, it isn't an issue with DeepFreeze anymore because I can make changes to the hard drive and they persist). Not sure what to do at this point?    Update 3: I ran the repair option and it couldn't repair, but it did point me to something. It says the error was due to a driver that was failing. I have a feeling it is my UPEK Fingerprint scanner. Symbols of derivatives Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$ I need help to solve following problem from Rudin's Mathematical analysis book:    Convergence of the series $\sum a_n$  implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n&gt;0$  I tried to construct a suitable convergence sequence $b_n$ such that $\sum b_n$ converges and $a_n \leq b_n$ but, I am not able to find such sequence $b_n$ .  Thanks for the help and sugestions. Modify systemd unit file without altering upstream unit file I have installed the pimd service by means of apt. This comes with an upstream systemd unit file (/lib/systemd/system/pimd.service).  I want the service to be restarted when for some reason it gets killed, hence I wish to add the line Restart = always in the unit file.   However, I don’t want to modify the upstream unit file.  Is there any workaround for this? How to align content of a div to the bottom Say I have the following CSS and HTML code:    #header {   height: 150px; } &lt;div id="header"&gt;   &lt;h1&gt;Header title&lt;/h1&gt;   Header content (one or multiple lines) &lt;/div&gt;     The header section is fixed height, but the header content may change.  I would like the content of the header to be vertically aligned to the bottom of the header section, so the last line of text "sticks" to the bottom of the header section.  So if there is only one line of text, it would be like:   ----------------------------- | Header title | | | | header content (resulting in one line) -----------------------------   And if there were three lines:   ----------------------------- | Header title | | header content (which is so | much stuff that it perfectly | spans over three lines) -----------------------------   How can this be done in CSS? Is there a constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$? Assuming the Axiom of Choice, every vector space has a basis, though it can be troublesome to show one explicitly. Is there any constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$, the vector space of real sequences? What is the distinction being made with regards to truth and falsity in these two sources? How do you set, clear, and toggle a single bit?