Prove that $\sqrt{47}$ is irrational number.   I know that a rational number is written as $\frac{p}{q}$ where $p$ &amp; $q$ are co-prime numbers. But I do not have any idea to prove it irrational number.   Thank you very much!
It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well.  On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$.  It seems that any positive integer has a square root that is either an integer or irrational number.        How do we prove that if $a \in  \mathbb N$, then $\sqrt a$ is an integer or an irrational number?      I also notice that I can modify the proof that $\sqrt{2}$ is irrational to prove that $\sqrt[3]{2}, \sqrt[4]{2}, \cdots$ are all irrational.  This suggests we can extend the previous result to other radicals.        Can we extend 1? That is, can we show that for any $a, b \in \mathbb{N}$, $a^{1/b}$ is either an integer or irrational?
Reference for general-topology In various articles it is mentioned that for secure communications, the recommended key sizes are 128-bit key size for symmetric encryption (which makes it $2^{128}$ possible keys?) and 2048-bit key size for asymmetric encryption ($2^{2048}$ possible keys?).  Why do they differ so greatly? It seems like I am missing a very big part of the equation. How do I calculate spoke length? Are unequal groups a problem for one-way ANOVA? I have data for three unequal groups: $N = 44$, $N = 354$ and $N = 347$. Is it possible to compare all three groups running a one-way ANOVA or is the first group too small? Story identification - "bless the beasts that bless us" I'm trying to trace what was probably a short story rather than a full novel from my childhood.  I think it starts in the present day (so the 80s or thereabouts) and there was some kind of alternate reality bleed which meant there were (i think) ghost boars or pigs, they were becoming more physical with time, and so problematic. The family escaped when the family dog led them to and through a portal which they couldn't perceive to a safer place.  In the end, they ended up in some kind of rural community who had a religion / custom that required them to say "Bless the beasts that bless us" but some folk shortened it to BTBTBU which irritated the girl who was the principle character.  I used to read a lot of kids short story sci-fi and fantasy including Nicolas Fisk but I don't know whether this was from a compilation of several authors or a single author's collection. Why is "cupboard" pronounced with a silent "p"? How can one make Firefox ignore my GTK theme entirely? Is there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do factorials and factorial roots (for lack of a better term)?   Here is an example: 5! = 120.  Is there a way I can work out the number that must be factorialed (??) to give the answer 120? Primes congruent to 1 mod 6 I came across a claim that I found interesting, but can't seem to prove for some reason. I have the feeling it should be easy  a prime $p$ can be written in the form $p = a^2 -ab +b^2$ for some $a,b\in\mathbb{Z}$ if and only if $p\equiv 1\bmod{6}$ Why does Stack Exchange use Imgur over Amazon S3 or Azure? What benefit does  offer over , , etc. for image hosting?   Is it the simplicity of integration? The cost model? What's better about using a "gateway"? I'm looking for an algorithm which when given a latitude and longitude pair and a vector translation in meters in Cartesian coordinates (x,y) would give me a new coordinate. Sort of like a reverse Haversine. I could also work with a distance and a heading transformation, but this would probably be slower and not as accurate. Ideally, the algorithm should be fast as I'm working on an embedded system. Accuracy is not critical, within 10 meters would be good. LibreOffice problem while openning a document My problem is that when I try to open one specific .odt document, I can't move the page or do anything except moving the mouse, which is useless. Also, the mouse, when I'm on the document, is freaking out switching between mouse and loader couple of times per second. It also becomes dark when I click on the document like if it's not responding. I am not talking about miracles, religious revelation, or artistic expression, but something more mundane. There is a lot of "empirical" evidence that the Riemann hypothesis is true, the scare quotes indicate that the source of the evidence is "mathematical intuition". In psychology there are mental states that no one sees, touches, hears, smells or tastes, qualia with their perceptual fullness that can not be conceptualized or directly communicated, and abilities/skills "know how" with the same problem. In linguistics Kripke's theory of reference relies on "modal intuitions" to decide the truth of counterfactuals. But an empirical counterfactual is an oxymoron, especially when it is embedded into an entire "possible world".  Freudian psychoanalysis and Husserlian phenomenology are attempts to deal with some of them systematically, but they are traditionally confined to the fringe of science at best. The mainstream approach is empirical second hand, but behaviorism in psychology and linguistics, and nominalism in mathematics, were not very productive.   According to the naturalized epistemology there is no a priori "first philosophy" of science, methodology is subject to the "tribunal of experience" and revision, just as the science itself. Should it apply to non-empirical experience? Sense empiricism has served science well for centuries, and was fully embraced by Quine.  takes him to task for it:     "Cognitive science is an empirical science working to unearth the mechanisms through which natural language constitutes itself. That account has had to recognize the indispensability of mental states, of beliefs, if it is ever to become adequate to the problem... There is still too much "first philosophy" in Quine. We must rescue naturalized epistemology from its own founder".    Note the use of "empirical" together with "mental states".  Questions: Does naturalized epistemology dictate that natural science should relax sense empiricism, and approach non-empirical phenomena partly first hand? If so, how can the scientific method be adapted to introspective/intuitive phenomena that are not readily reproducible, manipulable, measurable, and/or publicly accessible? What would play the selective role of empirical testing? Can there be non-empirical (more likely, not entirely empirical) natural science? What is the speed of sound in space? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver? How do you secure wood framing to a concrete basement floor? Cartesian Product of Two Countable Sets is Countable I have a following problem: when I plug-in my USB flash drive, it isn't shown in Windows Explorer or recognized by system at all. In device properties, it is shown as:     USB MEMORY BAR     There's no partitions on the drive, so I can't format it:    DISKPART:     Virtual Disk Service error:   There is no media in the device.   HP USB format tool:     There is no media in the specified device.   Any help will be greatly appreciated! Thank You! Comparing mixed-effects and fixed-effects models (testing significance of random effects) Given three variables, y and x, which are positive continuous, and z, which is categorical, I have two candidate models given by:  fit.me &lt;- lmer( y ~ 1 + x + ( 1 + x | factor(z) ) )   and  fit.fe &lt;- lm( y ~ 1 + x )   I hope to compare these models to determine which model is more appropriate. It seems to me that in some sense fit.fe is nested within fit.me. Typically, when this general scenario holds, a chi-squared test can be performed. In R, we can perform this test with the following command,  anova(fit.fe,fit.me)   When both models contain random-effects (generated by lmer from the lme4 package), the anova() command works fine. Owing to boundary parameters, it is normally advisable to test the resulting Chi-Square statistic via simulation, nonetheless, we can still use the statistic in the simulation procedure.  When both models contain only fixed-effects, this approach---and, the associated anova() command---work fine.  However, when one model contains random effects and the reduced model contains only fixed-effects, as in the above scenario, the anova() command doesn't work.  More specifically, I get the following error:   &gt; anova(fit.fe, fit.me)  Error: $ operator not defined for this S4 class   Is there anything wrong with using the Chi-Square approach from above (with simulation)? Or is this simply a problem of anova() not knowing how to deal with linear models generated by different functions?  In other words, would it be appropriate to manually generate the Chi-Square statistic derived from the models? If so, what are the appropriate degrees of freedom for comparing these models? By my reckoning:  $$ F = \frac{\left((SSE_{reduced}-SSE_{full})/(p-k)\right)}{\left((SSE_{full})/(n-p-1)\right)} \sim F_{p-k,n-p-1} $$  We are estimating two parameters in the fixed effects model (slope and intercept) and two more parameters (variance parameters for the random slope and random intercept) in the mixed-effects model. Typically, the intercept parameter isn't counted in the degrees of freedom computation, so that implies that $k=1$ and $p=k+2=3$; having said that I'm not sure if the variance parameters for the random-effects parameters should be included in the degrees of freedom computation; the variance estimates for fixed-effect parameters are not considered, but I believe that to be because the parameter estimates for fixed effects are assumed to be unknown constants whilst they are considered to be unknowable random variables for mixed effects. I would appreciate some assistance on this issue.  Finally, does anybody have a more appropriate (R-based) solution to comparing these models? Sobolev inequality If $f\in H^2(\mathbb R^2)$, I want to show that    $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$   For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb R^2} |f|\le\int_{\mathbb R^2} |\hat f(\xi)|d\xi = \int(1+|\xi|^2)^{-1}(1+|\xi|^2)|\hat f(\xi)|d\xi \le (\int (1+|\xi|^2)^{-2}d\xi)^2(\text {which is integrable in this case})||f||_{H^2}\to \text{is this correct?}$   But for 2, I am stucked. How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?