Let say $a_0, a_1, ..., a_k$ are a series of positive integers, with $k &gt; 0$, if one of the number has irrational square root, $\sqrt{a_n}$, such as $\sqrt{2}$, or $\sqrt{3}$, there is not way we can find a series of $a_0, ..., a_k$ to make  $$\sqrt{a_0}+ \sqrt{a_1} + \sqrt{a_2}+ ... + \sqrt{a_k}$$ is an integer.  Is this assumption correct? Is there a theorem for this? Thanks.
Is it true that all numbers of the form $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ are not integers, where $a_i,b_i$ are positive integers such that none of the $k$ terms is an integer? I think this must be a theorem somewhere, probably a fundamental theorem in number theory, but not sure what it would be called.
Suppose that sequences of real numbers satisfy:  \begin{align*}   a_n &amp;= \sum\limits_{r=0}^n \binom{n}{r} b_r \\ \end{align*}  Prove that:  \begin{align*}   (-1)^n b_n &amp;= \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s \\ \end{align*}  My work:  The $n=0$ case:  \begin{align*}   a_0 &amp;= b_0 \\   b_0 &amp;= a_0 \\ \end{align*}  Then $n=1$ case:  \begin{align*}   a_1 &amp;= \binom{1}{0} b_0 + \binom{1}{1} b_1 = b_0 + b_1 \\   -b_1 &amp;= \binom{1}{0} a_0 - \binom{1}{1} a_1 = a_0 - a_1 \\   b_1 &amp;= a_1 - a_0 = b_0 + b_1 - a_0 = b_1 \\ \end{align*}  The inductive step:  \begin{align*}   (-1)^{n+1} b_{n+1} &amp;= \sum\limits_{s=0}^{n+1} \binom{n+1}{s} (-1)^s a_s \\ \end{align*}  I'm not sure where to go from here. I tried substituting $a_s$ into that last equation and it didn't help. Any ideas? I'm running Apache2 in a docker container, and want to write nothing to the disk, writing logs to stdout and stderr. I've seen a few different ways to do this (,  ) but these seem like hacks. Is there no way to do this by default?   To be clear, I do not want to tail the log, since that will result in things being written to the disk in the container. Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ Comaximal ideals in a commutative ring Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? Is it possible to get rid of the bullets in the miniframes outer theme How can I find out when Windows was last restarted? How can I know when my computer running Windows 7 was last restarted?   I prefer a solution that doesn't involve searching the event log, but something like wmic or maybe cmd commands. 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. I completed the two contracts I was given (Scout and Heavy), and earned two of the 'Teufort' weapons for turning them in.  I expected to get some new contracts so I could continue to earn some more of the schmancy new weapons, but I can't seem to find out how to get more.  Do I need to buy the ticket again to unlock more contracts? If not, how are contracts assigned? How to retrieve the content/file in JSP? After these instructions in the Python interpreter one gets a window with a plot:  from matplotlib.pyplot import * plot([1,2,3]) show() # other code   Unfortunately, I don't know how to continue to interactively explore the figure created by show() while the program does further calculations.  Is it possible at all? Sometimes calculations are long and it would help if they would proceed during examination of intermediate results. On determinants and common divisors Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$ I inherited a project using SQL Server 2005.  Recently there were some performance problems and I started looking at the indexes, and found  _dta_index_survey_25_135059617__K1_K10_K19:     id ASC, sent_date ASC, group_id ASC   and  _dta_index_survey_21_364632442__K18_K1_K2_K9:     group_id ASC, id ASC, campaign_id ASC, sent_date ASC   which appears to be generated by some kind of automatic performance tuning tool.  I think SQL Server comes with something like that but I'm not familiar with it.  Anyway, id is the primary key, and what I'm trying to understand here is how it could possibly be beneficial to include the primary key within a secondary index... because you have to have the leftmost pieces of information to use an index and if you know id then you've got the row anyway.  Can someone explain why a tool would have generated these indexes and if there's any point in keeping them?  Just because the indexes are getting used doesn't mean they're a good design.  SQL Server is using the wrong indexes lately, and I'm trying to determine why. More specifically, queries against this table that ran in milliseconds a month ago are now taking multiple seconds.  The table is huge (4M rows), and growing steadily, but at the same pace it's been going at for years.  Almost every query joins across group_id or campaign_id.  Often both.  The application almost never specifically selects or sorts by the PK, but I could see how ID could be useful when comparing or merging lists of IDs within the database engine.  ... But id is also an identity column, and an index holds lists of row ids, so it still seems redundant to have the ID in the key when it's also in the index "buckets".  And the additional fields could still be covered by the index without being part of the key.  I discovered that (id, sent_date, group_id) hardly ever gets read.  Meanwhile (group_id, id, campaign_id, sent_date) is one of the hottest indexes.  Next I found that an index which I thought was pretty reasonable (campaign_id, sent_date, group_id, id) also has a index_columns_include of every other column in the table, and occupies more space than the primary clustered index on id!  (but it's also pretty hot...)  Surely I can replace it with a simple (campaign_id, sent_date, group_id)? How to save/restore Sublime Text 2 configs/plugins to migrate to another computer? Time complexity function in terms of theta notation sum = 0;  for (i = 0; i &lt; n; i++)    for (j = 1; j &lt; n^3; j = 3*j)                  sum++;   what is the time complexity (in $\Theta$-notation) in terms of $n$?  so far, this is what i've done: The running time is $\Theta(N^4)$  Because the for loop’s condition is depend on $n$ and is incremented by $1$, and  then inside running time is $\mathcal{O}(1)$. i feel i'm wrong here. any help would be appreciated. Where are icons stored? I want to change an icon of a specific theme. Where do themes save their icons? A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian? Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. Hydrostatic pressure? What what I understand, hydrostatic pressure is the "weight" of the water pushing against objects. But if this is true, why is hydrostatic pressure perpendicular to the surface it acts on instead of always going down?  For example, if you placed a book on a desk, the book's weight would push against the desk, but gravity is pulling it "down". But if you put another book beside this book the first book wouldn't apply any force on it. The weight exerted from the book always goes down.  Similarly, why doesn't hydrostatic pressure always go down? I understand that if you put a plate on the seabed, the weight of the water would push down on it, but if you just had a vertical plate standing on the bed, why would force push it from the sides?  To sum up, why does hydrostatic pressure act perpendicular to the object instead of always down? I would like to create a batch file, that will add registry key, not manually, but full automaticly by .BAT file.  goes about adding everything but not the key.  shows how to add everything, and how to add key manually, but it requires user interference, which I am trying to avoid.  Is there possibility to add KEY (non REG_*) to "folder?" in registry by one simple batch file? CPU usage is too high with Yosemite and an external monitor I use Yosemite 10.10.4 on my MacBook Pro (2012).  I used to work with an external monitor: Apple Thunderbolt 27" monitor.  The issue is that, according to Activity Monitor, the kernel_task process consumes more than 600% of CPU even when no tasks are running! The computer becomes totally unusable.   The workaround is to disconnect the external monitor, but I really need this secondary screen to work.  Why didn't Apple fix this issue in the latest Yosemite update?  Is there a fix for this?