I have tested for all primes less than  $40,000$ and it seems that, if $p$ is a prime then $p^2+26$ is not. I would like to see a proof or a counter-example.
I am not able to find an answer to the following question:     For which positive even integers $k$ is the integer   $$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
I am trying to change the page size in the QGIS 3.0 Composer but I am not able to see the Composition tab.  I have right clicked an empty space in the toolbar but did not see anything about composition.  I am using Windows 7.  I tried looking at the QGIS docs but they are still in testing. Allow users to put their e-mail in posts? Should I edit out the (thankfully obsfucated) e-mail address from this ?  Is this something we should allow on the site going forward?  I don't care about the "thank you" or "hello", but this might lead to an answer being sent by e-mail, which would mean it wouldn't be peer reviewed (for what that's worth), and the answer would be lost to the community. How to interpret Vgs(th) min/max in a MOSFET datasheet? This is a screenshot of IRF9640 MOSFET datasheet. Will you please tell me, what is meant by max and min value of Vgs threshold voltage. Also what they mean by test condition.  Whether the max and min values are only applicable for specified test condition? What will be the threshold voltage for a switching circuit and how the range varies?  My doubt is, for p channel mosfet to turn on Vgs &lt; Vgs(th). For IRF9640 in what volt it will turn on -3 or -5? When i used in my application Vs is 6v and Vg is 3v hence Vgs=-3v but it won't turned on, Why? It turned on when Vgs=-3.3v.  If we consider -2v as threshold, it should turn on when Vgs=-3v. Why they given a range for threshold voltage. Whether we have to test each mosfet to know their threshold. What i am pointing is that, if we have two IRF9640, will Vgs(th) is differ from each other? Subspaces of separable normed spaces Let $X$ be a separable normed space. Is it true that every subspace is separable? If it was Hilbert space I would take the dense set and then their projections.  It sounds trivial but I cannot prove or disprove it...  Thank you. Unresolved external symbol on static class members Very simply put:  I have a class that consists mostly of static public members, so I can group similar functions together that still have to be called from other classes/functions.  Anyway, I have defined two static unsigned char variables in my class public scope, when I try to modify these values in the same class' constructor, I am getting an "unresolved external symbol" error at compilation.  class test  { public:     static unsigned char X;     static unsigned char Y;      ...      test(); };  test::test()  {     X = 1;     Y = 2; }   I'm new to C++ so go easy on me. Why can't I do this? What's a better way to define individual link colours for a url in a macro? I've got a document where I'd like two colours of links - cyan for links in headers that since it's brighter, and navy blue for links in the main body which is more subtle/less distracting. I already tried using the xcolor package and changing the text colour with textcolor, like \textcolor{cyan}{\href{url.com}{link name}}, but it doesn't override whatever link or url colour I have defined in hypersetup.   So this is what I've done:   I've defined the colour navy:   \definecolor{navy}{HTML}{2F729C}    And also imported the hyperref package and defined commands to set the global link/url colours to cyan / navy.   \usepackage{hyperref}                         \hypersetup{colorlinks=true,linkcolor=navy,urlcolor=navy}                        \newcommand{\navylinks}{\hypersetup{linkcolor=navy,urlcolor=navy}}               \newcommand{\bluelinks}{\hypersetup{linkcolor=cyan,urlcolor=cyan}}     Then I've defined another command specifically for the header links, where I turn on cyan links globally, include the header text and link, and then turn on navy links again:   \newcommand{\headerlink}[1]{\bluelinks\href{#2}{#1}\navylinks}   My issue with this is that I feel like there could be a better way to do it than to set the global link colour every time I use a link macro; is there a way to define the link colour just for one link, like one would do with \textcolor{...}, that would be able to override hypersetup? Is there any way to have BibTeX or biblatex insert a bibliography without having citations in the body of the document itself? Maybe a way to hide the in-text citations, so that BibTeX still sees them and includes them in the bibliography, but does not print them in the body? How to permanently turn off online results I don't like getting online results when searching in my dashboard, so I went to System Settings &gt; Security &amp; Privacy &gt; Search, and I turned the setting off:    But when I restart my machine, the setting is always reset to:    So how can I permanently disable it without it turning on again after a restart?  Information Update:  I have had a look at the post that this has been marked as a possible duplicate of, and I don't think that my question really is a duplicate of it, as the other question is asking about how to set it globally, and I am not, I am only wanting to set it for one user, so the fact that it is not global is not the problem, the problem is that it is not sticking for that one user, and just resetting to its default state after a restart. In fact, I don't want it to be set globally, and the answers from the other question only really explain how to do that.    OS Information:  Description:    Ubuntu 15.04 Release:    15.04   Package Information:  unity:   Installed: 7.3.2+15.04.20150420-0ubuntu1   Candidate: 7.3.2+15.04.20150420-0ubuntu1   Version table:  *** 7.3.2+15.04.20150420-0ubuntu1 0         500 http://gb.archive.ubuntu.com/ubuntu/ vivid/main amd64 Packages         100 /var/lib/dpkg/status What are examples where a "naive bootstrap" fails? Is CAT5 cable good enough for RS-485 vs "true" RS-485 cable Find and replace text within a file using commands How can I find and replace specific words in a text file using command line? Does $P(A\cap B) + P(A\cap B^c) = P(A)$? Newman's "Natural proof"(Analytic) of Prime Number Theorem (1980) I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out.  1 . I am not clear, why in step (1)'s proof he says that from unique factorization and the absolute convergence of zeta we have... [here factorization makes sense but where did he use absolute convergence to prove (1).]  2 . Step 2 is clear but where will he use it (may be I will out later as i am still working on next steps.  3 . My major problem is to understand step 3rd's last part that i put it in the red box. please elaborate it. I didn't understand anything in this box.  I am still working on the next steps of the proofs. If i get any problem in them i will ask you.  Thank you so much in advance. Loop "Forgets" to Remove Some Items C# Interfaces. Implicit implementation versus Explicit implementation What are the differences in implementing interfaces implicitly and explicitly in C#?  When should you use implicit and when should you use explicit?  Are there any pros and/or cons to one or the other?    Microsoft's official guidelines (from first edition ) states that using explicit implementations are not recommended, since it gives the code unexpected behaviour.  I think this guideline is very valid in a pre-IoC-time, when you don't pass things around as interfaces.  Could anyone touch on that aspect as well? Why would a spinning space station create a centrifugal force on an astronaut rather than simply spinning around him/her? We often see films with spinning space station that create  by having the astronauts pulled outwards by centrifugal force.  I'd like to know if this would really happen, and if so, why is the following scenario not true:   Take an astronaut in open space. He doesn't move. Put a big open spinning cylinder around him - surely he still doesn't move. Close the cylinder. I still see no reason for him to be pulled outwards. I have four numeric variables. All of them are measures of soil quality. Higher the variable, higher the quality. The range for all of them is different:  Var1 from 1 to 10  Var2 from 1000 to 2000  Var3 from 150 to 300  Var4 from 0 to 5  I need to combine four variables into single soil quality score which will successfully rank order.  My idea is very simple. Standardize all four variables, sum them up and whatever you get is the score which should rank-order. Do you see any problem with applying this approach. Is there any other (better) approach that you would recommend?  Thanks  Edit:  Thanks guys. A lot of discussion went into "domain expertise"... Agriculture stuff... Whereas I expected more stats-talk. In terms of technique that I will be using... It will probably be simple z-score summation + logistic regression as an experiment. Because vast majority of samples has poor quality 90% I'm going to combine 3 quality categories into one and basically have binary problem (somequality vs no-quality). I kill two birds with one stone. I increase my sample in terms of event rate and I make a use of experts by getting them to clasify my samples. Expert classified samples will then be used to fit log-reg model to maximize level of concordance / discordance with the experts.... How does that sound to you? Infinite Product $\prod\limits_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$ From Berkeley Problems in Mathematics, Spring 1999, .  Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\ge 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in L$ be a zero of $f$. Give that $[L:\mathbb{Q}]=n!$, prove that $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$.   (BELOW IS NONSENSE, SHOULD BE IGNORED)  Following Gerry Myerson's advice, since $L$ is the splitting field of $f$, $L$ must be Galois over $\mathbb{Q}$. Thus the Galois group for $[L:\mathbb{Q}]=S_{n}$. Gerry now asserts that either $\mathbb{Q}(\alpha^{4})$ is of degree 2 over rationals or it must be $\mathbb{Q}$. But assuming this one can rule out the case $\mathbb{Q}$, since this would imply $\alpha$ is the root of a 4th or 2nd degree polynomial over $\mathbb{Q}$. But we know $n\ge 3$. So $\alpha$ must be the root of a 4th degree polynomial. We know $S_{n}$ has normal subgroups $A_{n}$ when $n\not=4$ and $V,A_{4}$ when $n=4$. I do not know how to proceed any further.   Now assuming $\mathbb{Q}(\alpha^{4})$ is a degree 2 abelian extension over $\mathbb{Q}$. We should have a chain of normal extensions $\mathbb{Q}(\alpha)\supset \mathbb{Q}(\alpha^{2})\supset \mathbb{Q}$. This would imply $S_{n}$ has a normal subgroup which has a normal subgroup. But we know $A_{n}$ for $n\ge 5$ are simple. So the only possibility is $\mathbb{Q}(\alpha)$ has a Galois group isomorphic to $V$. In this case $n=4$ as well. I can only proceed to here. Filter "Hot Network Questions" by excluding sites