primes of the form $4k+3$ and sums of squares
Can an integer of the form $4n+3$ written as a sum of two squares?
Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know under what conditions the identities have to be equal. I know they must be equal if $S$ has no zero divisors, since for every $r\in R$ we have $(1_S-1_R)r=0$  In other words: Is the category of unital rings with identity-preserving morphisms a full subcategory of the category of rings (where morphism are only required to respect addition and multiplication)? I would like to be able to render several files one after the other, when I am not using my computer, e.g. at night. Is there a way of doing this? What's the projective limit of these polynomial rings ? Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m&lt; n)$.   Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ? Order of a homomorphism of groups How can I understand and prove the "sum and difference formulas" in trigonometry? Slipstream Windows 7 + Service Pack 1 How should we deal with duplicate answers? Duplicate questions are "solved" &#8212; it's clear what to do with them, there're tools to deal with them... but what happens when two users post the same answer? This happens a fair bit, particularly on questions with simple answers ("Click this button"-type ones).  Do we upvote both answers, since they are both correct? Do we upvote the first one (although if the answers were posted within a few minutes of each other, after an hour it's impossible to tell as they both show up as "answered 1 hour ago")? Do we upvote the currently highest voted one? Do we upvote one and downvote the other? Do we flag one (seems a little excessive)? Something else? How can I view the contents of tar.gz file without extracting from the command-line? I want to see the contents (list of files and folders) of an archive, for example a  tar.gz file without extracting it.  Are there any methods for doing that? I have a list of integers, List&lt;Integer&gt; and I'd like to convert all the integer objects into Strings, thus finishing up with a new List&lt;String&gt;.  Naturally, I could create a new List&lt;String&gt; and loop through the list calling String.valueOf() for each integer, but I was wondering if there was a better (read: more automatic) way of doing it? Calculate $\lim_{n\rightarrow +\infty}\binom{2n} n$ This  gives this comparison chart for the four Windows 8 Editions (Core is usually just called Windows 8). I notice specifically, about half-way down, the max physical processors is listed as 1 for "Core" and 2 for all other.  In the second image i see physical processors listed as 8 in a Windows 8 Task Manager.  My question is, what is the limit then? Can anyone verify the article or the screenshot?  Was this a mock-up? Another Edition? (Server perhaps?)      To clarify, i am looking for links to official microsoft.com articles detailing these specifications.  I have been unable to find any. Move incoming Outlook emails to folder containing previous conversation I file email conversations in various folders. But every new email in that conversation goes to inbox, so I have to manually move it to the folder that contains the rest of the conversation. Is there a way for Outlook to do this automatically with all new emails?  What I want: Outlook will do the following automatically as long as it's running:   For all unread emails in inbox... ...if email is not the first in the conversation, search for folders where that conversation is stored. If found, move the new unread email to the first* folder it finds (except for Inbox, Trash, or Sent). Optionally: If there is more than one folder containing messages in the same conversation, then do nothing.   **If there is more than one folder containing messages in the same conversation, then do nothing*. (This extra rule is not necessary if it complicates the solution too much)  Thanks! How to combine Bump Maps? I'm trying to add a bump map with logo to noise texture Bump map.  I have a bump map created with Noise texture and then on top of that I want to add my bump logo (which also has it's own color on it.)   Is it possible within one material? Prove that the set of palindromes are not regular languages using the pumping lemma. Firstly I pick a language $xyz$ where $x = \epsilon$, $y = (abb)^{k}$, $z = (bba)^{k}$ where $|y| \ge$ the number of states in the automaton representing my language. Then $xyz = (abb)^k(bba)^k$ is a palindrome.  Now I split $y$ into $uvw$, where $v \ne \epsilon$ contains a state visited more  than once. If the language is regular then $xuv^iwz$ is within the language $\forall i \ge 0$.  I choose $i = 2$, then $uv^2w = (abb)^n$ where $n &gt; k$.  Then $xuv^2wz = (abb)^n(bba)^k$ where $n &gt; k$, which is not a palindrome.   Firstly, is my reasoning for this particular proof correct?  I'm unsure about how to properly apply the pumping lemma to this problem. And secondly the question asks me to state a proof for palindromes in general, whereas I only attempted to prove it for a single case. Is there an example I can choose which proves the conjecture for all palindromes? I assume not, seeing as I can't think of how that could be represented using regular expressions. Independence of disjoint events On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?) I got a new laptop; it came with an existing Windows installation, but I wanted to migrate my current backup to the new machine. I dd-ed the new machine's partitions to an external drive:  /dev/nvme0n1p1 /dev/nvme0n1p2 /dev/nvme0n1p3 /dev/nvme0n1p4   and afterwards move my current partitions to the new machine. What followed was a medium-sized nightmare. The new machine would boot, but the wireless network drivers were not recognized and later found out (to the best of my knowledge) they couldn't be installed. I tried:   booting from the console and running:  sfc /scannow chkdsk C: /f /r /x   both seemed to have completed successfully, but now the machine would get stuck in a startup repair loop. I tried resetting the machine (whilst keeping my files), but this failed halfway through. now the machine's still stuck in the loop and decided to try and repair it externally using a bootable Windows&nbsp;10 . It's likely that I'll need the new machine's Windows&nbsp;10 key. I'm not sure if it's stored in the BIOS (it came with Windows&nbsp;10  preinstalled) or somewhere on the partitions.   How can I get the key from the partitions (if it's there at all) while booting from a Linux image? Error AL lib: UpdateDeviceParams: Failed to set 44100hz, got 48000hz instead. What can I do? I have Win 10 and when I open Blender I have the error:     AL lib: UpdateDeviceParams: Failed to set 44100hz, got 48000hz instead Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?    Addendum by LePressentiment on 11/29/2013: (in the interest of  and averting a duplicate)  What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally. Meaning of Sun Lamp Size?