How can I prove that all prime numbers have irrational square roots?  My work so far: suppose that a prime p = a*a then p is divisible by a. Contradiction. Did I begin correctly? How to continue?
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?
Google Analytics configuration setting for Drupal multisite (domain and submain) I am working on a Drupal mulsite project (using single database) and using Google Analytics module for this.  I have follow the below steps for GA configuration:   Created a domain name example1.com and a sub-domain name example2.com. Create an account on GA and create property for both of this domain name in GA account. Both property give me two different Tracking ID (same as below screenshot).      Into the configuration setting of GA from drupal admin section I have added Web Property ID of example1.com and into domain setting added both the domain name example1.com and example2.com into the section of List of top-level domains (a same as attached screenshot).      Now this configuration is only working for emample1.com, not for example2.com. So can anyone please tell me how can I use Web Property ID of domain exmaple2.php to get the result? I consistently have more than one terminal open. Anywhere from two to ten, doing various bits and bobs. Now let's say I restart and open up another set of terminals. Some remember certain things, some forget.  I want a history that:   Remembers everything from every terminal Is instantly accessible from every terminal (eg if I ls in one, switch to another already-running terminal and then press up, ls shows up) Doesn't forget command if there are spaces at the front of the command.   Anything I can do to make bash work more like that? Vim-latex adds extra brackets when typing empty brackets It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. I would like to be able to execute a SOQL query that will allow me to see which users have access to a certain record type for a certain object.  For example, if I have a case record type called Phone Team, I would like to make a query that returns the names of all the users that have been assigned the Phone Team record type, as can be seen in the native user interface in a profile's Object Settings page.  So far I have managed to get a query that can get the names of all the users that have, for example, read access to the case object:  SELECT Assignee.Name FROM PermissionSetAssignment      WHERE PermissionSetId IN          (SELECT ParentId FROM ObjectPermissions WHERE SObjectType = 'Case'               AND PermissionsRead = true AND Parent.IsOwnedByProfile = true)   However, there doesn't seem to be any way to access record type assignment in the same way that you can object permissions.  I see from  that the info can be obtained via the metadata API, but I would much prefer to do it by SOQL if possible.  Thanks What is a terminal? How do I open a terminal window? How do I use the terminal? Package repository down, how to relocate? I would like to use the function of TeXnicCenter 1.0, where is automatically downloads missing packages used. The problem is, that the repository it wants to download from does not exist anymore (the one on piotrkosoft.net).  How do I get TeXnicCenter to try with another repository? What is the best way of defining white noise process so it is intuitive and easy to understand? Let $x$ and $y$ be integers not congruent to $0$ modulo $p$ where $p$ is a prime. Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$.   I thought about proving this algebraically and by contradiction. So we say $x^2+y^2 = pz$ and have to show this is impossible. Blender 2.80 system Req + GPU req Can someone please let me know the minimum system requirements and GPU requirement to run blender 2.80 and eevee smoothly?   Also please let me know the optimum system requirements for the purpose of    environment modeling Game development charachter modelling and animation   Also if someone can suggest me a good desktop/laptop for personal use that is compatible with blender 2.80 in a low budget.  Any help will be much appreciated.  Thanks in advance! 750Gig Hard Drive shows full with only 315Gigs used I have a laptop with win7 installed. I have now made a 60gb partition which I want to install ubuntu into. The question I have, before I do the installation, is how large each of the root, swap, and home partition should be? I have read some place that root could be as small as 8GB, but isn't that too small? Since I guess beside ubuntu all the softwares installed will reside there as well? And I think I'm going to set my swap to be 2GB large.  My main concern is how large should the root partition should be. I'm mainly going to use ubuntu for programming and browse the web. Suppose I have a field that contain the following data:   ========================== |  Email Address         | ========================== | Jack@wonderland.com    | | Peter@nobodyland.com   | | Jill@wonderland.com    | ==========================   Is there a calculated formula in SharePoint that will give me the following result? (It will help me to retrieve the sub-string from the Email Address column and give me the answer as shown in the UserName and Domain column):   =========================================================== |  Email Address         |  UserName  |      Domain       | =========================================================== | Jack@wonderland.com    |   Jack     |   wonderland.com  | | Peter@nobodyland.com   |   Peter    |   nobodyland.com  | | Jill@wonderland.com    |   Jill     |   wonderland.com  | =========================================================== How did Newton discover his second law? I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon?   According to , the  reads:     Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.   Translated as:      Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.   My question is how did Newton come to this conclusion? I get that he knew from  the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this? Find the integer solutions: $$a·b^5+3=x^3,a^5·b+3=y^3$$ This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$.  Thanks in advance! Why comment reply auto-completion feature (@) sometimes doesn't work? I often need to make banner sets that consist of anywhere from 6-14 different sized banners. A lot of these banners have the same text (which I put in smart objects to work faster) and the same or different images. The placement and sizes of these elements differ according to the banner size though.  I find the process of 'copying' the content from one artboard to another to go a bit slower than I'd like it to. I can't simply copy/paste, I have to duplicate the layer and then drag it to the right artboard in the layer panel. Hoping I have closed all my folders or else I have to drag miles upwards ;)   There has to be a better way! Why can't I simply copy paste a layer anymore? I'm pretty sure this used to be the case in older versions of photoshop. I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup of order $n$"). This is an exercise from a book, and it is in the section on finite abelian groups, so I know I have to use the fundamental theorem of finite abelian groups. I have come up with a proof, but it seems a bit messy, and I'm not entirely sure if it's correct. It is given below.     Let the order of $G$ be $m$ = $p_1^{\alpha_1} \ldots p_k^{\alpha_k}$. It is known that $G$ is a direct product of $p$-groups, say:      $$G = G_1 \times \ldots \times G_k$$      where each $G_i$ is a $p_i$-group. By the fundamental theorem of finite abelian groups, each $G_i$ is isomorphic to a direct product of cyclic groups of the form      $$\mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}},$$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$. Now if $n$ divides $m$, then we must have      $$n = p_1^{\gamma_1} \ldots p_k^{\gamma_k}$$      for some $\gamma_1, \ldots, \gamma_k$ with $0 \leq \gamma_i \leq \alpha_i$.      Claim: Each $G_i$ has a subgroup of order $p_i^{\gamma_i}$      Proof: As above, we have that       $$ G_i \cong \mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}} $$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$.      Now since $0 \leq \gamma_i \leq \alpha_i$, we can find   $l$ numbers $\delta_1, \ldots , \delta_l$ such that $\gamma_i = \sum_{j=1}^l \delta_j$, and $0 \leq \delta_j \leq \beta_j$. (This choice of numbers is not necessarily unique).      Then $p_i^{\delta_j} | p_i^{\beta_j}$ for each $j = 1, \ldots , l$. Hence, for each factor $\mathbb{Z}_{{p_i}^{\beta_j}}$, there exists a subgroup of order $p_i^{\delta_j}$, namely $\mathbb{Z}_{{p_i}^{\delta_j}}$ (using the fact that the converse of Lagrange's theorem is true for finite cyclic groups). Taking the direct product of each of these subgroups, we get a new subgroup $G_i'$ of $G_i$:      $$G_i' \cong \mathbb{Z}_{{p_i}^{\delta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\delta_l}}$$      The order of this subgroup is $p_i^{\delta_1} \times \ldots \times p_i^{\delta_l} = p_i^{\delta_1 + \ldots + \delta_l} = p_i^{\gamma_i} $. So we have found a subgroup of $G_i$ of order $p_i^{\gamma_i}$, as required.      So each factor $G_i$ in the product $G = G_1 \times \ldots \times G_k$ has a subgroup $G_i'$ of order $p_i^{\gamma_i}$.       Therefore, $G$ has a subgroup   $$G_1' \times G_2' \times \ldots \times G_k'$$       of order $p_1^{\gamma_i}...p_k^{\gamma_k} = n$, which completes the proof.   I have two questions about this: firstly, does this proof seem to work? Secondly, is there a way to make the proof more concise (e.g. a way to prove the statement without using all these indices)? There's already a question like this  so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.   Is there a way to derive  from the Lorentz' Force law or just from Maxwell's Equations?   The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.   Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?  I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.   In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow? What's in a name: hyperparameters So in a normal distribution, we have two parameters: mean $\mu$ and variance $\sigma^2$. In the book Pattern Recognition and Machine Learning, there suddenly appears a hyperparameter $\lambda$ in the  regularization terms of the error function.  What are hyperparameters? Why are they named as such? And how are they intuitively different from parameters in general?