Statement : A prime $p$ is of the form : $3 \pmod 4$. Show that if $p \mid a^2+b^2$, then $p \mid a$ and $p\mid b$.   I don't know exactly how to proceed. I got the following proof from somewhere:     Assuming that $p\mid a^2+b^2$ and $p\nmid a$, we get that $\exists k \ni ak\equiv 1 \pmod p \implies (ak)^2+(bk)^2\equiv k^2(a^2+b^2)\equiv 0 \pmod p \implies (bk)^2 \equiv -1 \pmod p$. And this is a contradiction.   Please explain me why $(bk)^2\equiv -1 \pmod p$ leads to a contradiction.
I came to this conclusion: the primes that can't be written as the sum of two squares (lets use$p\prime$ to denote) can be written as ${p}\prime \equiv 3 \;(\bmod\; 4)$.  I need a simple proof to show this and am stuck. I would appreciate guidance!
System.ListException: Before Insert or Upsert list must not have two identically equal elements - Deep Dive Why won't QGIS recognize my GPS? Proof: $\cos^p (\theta) \le \cos(p\theta)$ I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty bucks is a lot of money when you don't have a job). So I took my phone out and started taking as many pictures as I could. Unfortunately, I didn't take any pictures of the solutions!  Enough backstory. Time for math. The problem is as follows:  Prove that $\cos^p( \theta) \le \cos(p \theta)$ if $0\le\theta\le\frac\pi2$ and $0\le p\le 1$.  I tried using the series expansion for cosine, but that seemed to be a dead end. Then I tried using Euler's theorem, but I got stuck. Then I got distracted and started to think about other cosine identities. For example, $$\cos\left(\frac\theta2\right)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}.$$ Then what is $\cos(\frac\theta3)$ equal to? I tried to figure it out then I realized that I needed to find the root of a cubic polynomial. Then I realized that I'm better at staying focused than finding the roots of cubic polynomials.  Anyways, a proof would be nice here. I really appreciate any hints or answers. I apologize for my digressions! Would a Bayesian admit that there is one fixed parameter value? In Bayesian data analysis, parameters are treated as random variables. This stems from the Bayesian subjective conceptualization of probability. But do Bayesians theoretically acknowledge that there is one true fixed parameter value out in the 'real world?'  It seems like the obvious answer is 'yes', because then trying to estimate the parameter would almost be nonsensical. An academic citation for this answer would be greatly appreciated. Finding the solution of the congruence $4x\equiv16\pmod{26}$ Solve the congruence   $$4x\equiv16\mod{26}.$$  How do I find the solution to this? I have tried by the euclidean algorithm but the gcd is not $1$ so it doesn't work.  $$\begin{align} 26&amp;=&amp;6&amp;\times4&amp;+2\\ 4&amp;=&amp;2&amp;\times2&amp;+0 \end{align}$$  Note: I understand we can see that $4$ and $17$ are solutions but how would I work this out algorithmically? Find the minimal polynomial for $\sqrt[3]{2} + \sqrt[3]{4}$ over $\mathbb{Q}$  I havent covered galois theory, this is an exercise from the chapter algebraic extensions in gallian.  I can see that $\sqrt[3]{2} + \sqrt[3]{4}$ $\in \mathbb{Q}(2^{1/3})$, after this I am clueless.  Need hints will finish the proof. Computer doesn't boot up, shows PXE error and/or "operating system not found", "no bootable device", "insert boot media", or other similar error Regex matches but shouldn't Questions about insects eating flour and living in the cracks of a wooden cutting boards I have a large wooden cutting board which is used in my kitchen mainly for preparing egg pasta. The wood type is Populus (poplar or aspen or cottonwood).  The main ingredients used with the cutting board are: flour (mainly soft wheat flour), hens' eggs, white sugar, butter, and baking powder.  After the food preparation, the cutting board is cleaned by scraping away the ingredients remains.  The board has some cracks and some small unknown insects are living inside the cracks.   The board has also a lot of holes due to woodboring beetles but I think the woodworms operated long time ago and they no longer live in the cutting board.  The unknown insect has an approximate length of 1 mm (along the direction of their movements) and a width of 0.3 mm. The unknown insects do not have wings, they just walk. I am not able to see how many pairs of legs they have.  The color of the insect is pale brown.  The cutting board was free from these unknown insects some months ago but then they appeared.  My questions:   What insects are they? Are these insects poisonous to humans? How can I get rid of them?   Update  The unknown insects could be Acarus siro, I will try to get a picture of the insects.  Update 2  Here is a picture. The bottom part of the image is a one Euro coin, the top part of the image is a caliper (each bright area is one mm tick).   Some more pictures: Is there a bijection of the natural numbers which swaps $\frac{1}{n}$-summable subsets with $\frac{1}{\sqrt{n}}$-summable subsets? Let me start with a precise statement of the question.  For a subset $A\subseteq \mathbb{N}$ and a series of positive real numbers $\sum_{n=0}^\infty a_n$, I'll use the notation $\sum_A a_n$ as a shorthand for $\sum_{n\in A} a_n$.  Is there a bijection $f:\mathbb{N}\rightarrow \mathbb{N}$ with the property that for every $A\subseteq \mathbb{N}$, $\sum_A \frac{1}{n}$ converges iff $\sum_{f(A)} \frac{1}{\sqrt{n}}$ converges?  Background: Fix a series of positive terms $\sum_{n=0}^\infty a_n$.  Given a subset $A\subseteq \mathbb{N}$, call it $a_n$-small if $\sum_A a_n$ converges.  The following proposition is easy to prove:  Proposition:  The set $\{A\subseteq \mathbb{N}: A\text{ is } a_n\text{-small}\}\cup \{\mathbb{N}\}$ is a topology of closed sets on $\mathbb{N}$.  I will use the notation $(\mathbb{N},a_n)$ to refer to this topology.  Then one can ask how topological properties of $(\mathbb{N},a_n)$ are related to series properties of $\sum a_n$.  For example, I can show:  Proposition:  The following are equivalent.  $\sum a_n$ converges.  $(\mathbb{N},a_n)$ is discrete.  $(\mathbb{N},a_n)$ is disconnected.  $(\mathbb{N},a_n)$ is Hausdorff    And there are other nice things.  For example, $(\mathbb{N},a_n)$ is compact iff $(\mathbb{N},a_n)$ is cofinite iff $\liminf a_n &gt; 0$. With this language, my question can be reformulated as...  Are $\left(\mathbb{N}, \frac{1}{n}\right)$ and $\left(\mathbb{N}, \frac{1}{\sqrt{n}}\right)$ homeomorphic?  I have made very little progress on this.  Of course, the identity function $i:(\mathbb{N}, \frac{1}{\sqrt{n}})\rightarrow (\mathbb{N}, \frac{1}{n})$ is a continuous bijection, but the inverse map is not continuous.  Also, if there is such a bijection, there there is such a bijection with agrees with $i$ on any preassigned finite set. Edit  I thought I add in a slightly suprising (to me, at least) example when things work out to be homeomorphic. Begin with a convergent series $\sum c_n$. and divergent series $\sum a_n$ with $\lim a_n = 0$.  Create a new series $b_n$ using all the terms of $c_n$ and $a_n$ (in whatever order you wish).  Then $(\mathbb{N}, a_n)$ and $(\mathbb{N}, b_n)$ are homeomorphic.  The idea is that since $\lim a_n = 0$, there is a convergent infinite subseries $\sum_A a_n$.  Then we use a bijection which with $A$ and $A\cup \{\text{indices of }c_n\}$ to &quot;squeeze&quot; the $c_n$ in without changing the topology.  Of course, I am glossing over many details, but I can include them if desired. How to avoid messages from applications which I ran with a &? I ran dolphin like this:  dolphin . &amp;   I see messages in the console like these:  QPixmap::scaled: Pixmap is a null pixmap   How can I keep the console clean or silent? $(12)$ and $(123\dots n)$ are generators of $S_n$ What iron-clad reason did Dumbledore have for believing Snape was on his side What iron-clad reason did Dumbledore have for believing Snape was on his side? Maybe he was a good actor and a double agent. Was it just that he loved (or pretended to) Lily or he used legilimency? Making overview with linked detailed maps in QGIS? i want to create 4-5 detailed maps, and then an overview map that has rectangles that markes the locations of the detailed maps. Best case would be if the rectangles were links to the detailed map.   I have managed to make the overview and the rectangles, and think I know how to make the detailed maps one by one. The rectangles are drawn manually on the overview map. All layers that I plan on using are in the same QGIS project.  The final product that I hope to create would be the overview map on the first page, followed by the detailed maps. The overview would have links to the detailed maps, click on the rectangles to open the right detailed map.  Is there a way to link the detailed maps to the overview in QGIS 2.8 print composer?  I'm still not sure if I'm doing something wrong, when I try to create the overview. I want it to show the extents of all of my detailed maps.   Is this possible? Can someone give me some example code that creates a surface with a transparent background in pygame? Apparently I inadvertently irked a moderator by creating a tag.  The moderator told me that my tag was "meta" and therefore not appropriate.  Having been a SO user for a long time (and a lurker on MetaSO), it was not clear to me what makes my tag "meta" and why meta tags would be inappropriate.  Many of the questions I see on SO aren't really about the topics in the title or the existing tag list.  For example, , isn't really about C++ or SetWindowText (or the Windows API as tagged).  It's simply somebody who had a bug, couldn't find it, and tried to outsource the debugging.  This seems to be a common category of question on SO.  So I started tagged questions like this with the label "findmybug".  Lots of frequent SO contributors regularly answer questions in this category rather than voting to close them as "Too Localized".  I took that as a cue that I should also try to answer rather than voting to close.  But leaving these questions open and tagged with "c++" and "winapi" even though the question isn't really about C++ or the Windows API seemed misleading.  Someone searching for answers on using SetWindowText from C++ probably isn't making exactly the same mistake and won't find that entry useful.  (In the end, the sample question was closed as "Too Localized", but this doesn't seem common for these types of questions.)  The Tags page says:     A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question.   It seems to me that the "findmybug" tag does group these similar questions together.  So from that perspective, it seems to fit.  Lots of users seem to enjoy answering these types of questions, so it makes sense they'd want a way to find them.  If you want to suggest that all the questions I would tag as "findmybug" should be closed as "Too Localized", that's fine.  But I'm trying to ask the more general question:  What makes a tag meta, and are meta tags inappropriate? How do you test that a Python function throws an exception? How does one write a unittest that fails only if a function doesn't throw an expected exception? Why is there one P in "hoping" and two P's in "hopping"? Hoping is the present participle of hope. hopping is the present participle of hop. Hoping has only one p while hopping has two. What difference does it make? Why is hoping not hopping?  Edit: I changed mop to hop because of a comment by Kat.  Kat: Why would you compare hope to mop instead of hope to hop (in which case I think your answer will become obvious)? How can energy be useful when it is 'abstract'? The topic which haunted me for two years until I gave up on it. But now I am doing engineering and this topic suddenly popped out of my textbook from nowhere. I seriously need to understand this topic because I wasn't able to do so in the past. I have read many book on 'energy' and it came to nothing. Maybe just because the many books I have read all said 'energy' is something which doesn't 'exist'! It is 'abstract'! It is just a number which represents the state or orientation of a system. But then I see so many examples that 'uses' energy to do 'work'. But the question here is that if anything doesn't exist in this universe, then how can the same thing be used to do something that exist?  My problem is that I didn't understood the topic 'energy' and all the other topics related to it (work, power etc.) As a follow up to my other , I would like to know whether the following is possible:  I have $20,000 in my brokerage account The price of a call option with $300 strike price is $5.00 (1 contract = 100 shares) I buy 20 contracts of these $300 calls (cost is 20 * 5 * 100 = $10,000) The price of the underlying goes up, making my option in-the-money  I decide to exercise my options. Exercising my calls means that I buy 20 * 100 = 2,000 shares of the underlying at $300 per share. However, I only have $10,000 left in my brokerage account and am unable to pay for the shares (2,000 * 300 = $600,000). What now? How does this work? Do brokers recognize that my position is profitable, and buy and sell the underlying for me as soon as I exercise my calls? In other words, do I need to worry about being able to pay for the shares of the underlying if I decide to exercise my options? Can I just take my profit without bothering with the shares at all?