I am a high school student interested in studying mathematics at university. I've been doing some independent study and have found the Gaussian Integers $\mathbb{Z}[i]$ particularly interesting. I was thinking about primes in $\mathbb{Z}[i]$, and came across the following results: First, considering which primes $p$ in $\mathbb{Z}$ remain prime prime in $\mathbb{Z}[i]$, I think the following is true:  If $p \equiv 3 \bmod{4}$, then it remains prime in $\mathbb{Z}[i]$. If $p \equiv 1 \bmod{4}$, then $p$ is not prime in $\mathbb{Z}[i]$. This is because $p$ can be written as a sum of two squares: $p = a^2 +b^2$, and so factorizes in $\mathbb{Z}[i]$ as: $p = (a+ib)(a-ib)$, where neither $a+ib$ nor $a-ib$ are units (the latter is true because $p \neq 1$).  However, I realized that this is not the end of the story, as there could be primes in $\mathbb{Z}[i]$ of the form $a +ib$, where neither a nor b equals zero. In this case, the only result I could find was that if $a^2 + b^2$ is prime (e.g. in the case of $1+i$) then $a+ib$ is prime in $Z[i]$. But this does not determine all the primes in $\mathbb{Z}[i]$. My question is: Is there a known necessary and sufficient criterion for a Gaussian Integer $a+ib$ to be prime in $\mathbb{Z}[i]$? If not, are there any additional primality tests known in $\mathbb{Z}[i]$? Any help and guidance would be greatly appreciated.
I am trying to prove the following: Let $R=\mathbb{Z}[i]$, $a, b \in \mathbb{Z}, ab \neq 0$. Show that $a+bi$ is an irreducible element in $R$ if and only if $a^2 + b^2 ∈ \mathbb{Z}$ is prime. Since $R$ is a Euclidean domain, irreducible elements are prime. I think this is relevant, but am stuck on where to go from here.
Cauchy-Product of non-absolutely convergent series While grading some basic coursework on analysis, I read an argument, that a Cauchy product of two series that converge but not absolutely can never converge i.e. if $\sum a_n$, $\sum b_n$ converge but not absolutely, the series $\sum c_n$ with $$c_n= \sum_{k=0}^n a_{n-k}b_k$$ diverges.  Although we didn't have any theorem in the course stating something like this, it made me wonder if it was true. Use of 100K ohm resistor along with 0.1uF capacitor? In the circuit diagram below, why is there a 100K&Omega; resistor (NOT R2) connected to the capacitor? To my understanding the capacitor-resistor act as a high pass filter to block the DC offset of the microphone, but since only the capacitor blocks DC why is the 100k resistor used? According to the author of the video (link below) he said, the 100k is used "not to overload the microphone's un-amplified output". I don't get this part.  Also, can only a capacitor be used in this circuit or any other circuit without the 100k resistor? I'm currently making a Minesweeper in C#. I've already done the basics (generating minefield, rules and ui) but I don't know how I should implement when the user clicks a cell and the cell has no mines nearby and clicks nearby cells with also no mines nearby, like this:    And after click    This is my current code on this:  if (MineField(x,y) == false) // If the clicked cell isn't a mine {     if (GetNeighbours(x,y) == 0) // GetNeighbours returns the amount of mines in the closest 8 cells     {         // To be honest, no idea...     } } I want to automatically split all lines in a QGIS layer at their vertices (). I've done a Google search and found , which is quite close but the workflow is complicated. Because I'll have to train my co-workers (who are unfamiliar with GIS) to use this tool, I want something as simple as a plugin or something that doesn't require GRASS. Since it's been a long time from 2010 - the time of , I wonder if there is any better way to achieve the same result? Simple Hadamard Circuit gives incorrect results? Is it a wise-decision to buy full-frame lenses for use on an APS-C body? I have a Canon 550D which I've been using for around a year now. I really love it and have learned a great deal with this and the 18 - 135mm kit lens. But now with experience I feel the need to get some better glass. I won't be buying a new body any time soon because I figure investing in some good glass is more important.   As I have looked around, I see quite a few good used L lenses I can get in ebay for good bargain prices. But I know that the EF lenses don't have the same focal length in the cropped sensors. Is it a wise decision to buy these EF lenses or buy the cropped sensor alternatives?  I have the following lenses in mind that I am looking to buy down the line from the canon range over the next several months.  16-35mm f/2.8L, 24-70mm f/2.8L, 70-200 f/4L.  Are these apt choices for cropped sensors? I know most pros with FFs use these but I am a bit skeptic that they will serve the same purpose for me. If so what are the alternatives for this that have almost as good image quality? How can I tell if an ETF is expensive? For stocks, I can easily look at the price/earnings ratio, and, according to The Intelligent Investor, stand clear of those with a ratio higher than 20.  How can I tell if the current price of an ETF is a good price to buy it? Are there such simple metrics? What are the ways to solve cubic equations? Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very helpful for students and learners. Is is possible to skip the confirmation step while adding a repository? When I add a repository by running  sudo add-apt-repository ppa:ubuntu-x-swat/x-updates   It outputs this dialog:  Updated versions of X.org drivers, libraries, etc. for Ubuntu.  This PPA is for stable upstream releases of X.org components.  If you're looking for something even more bleeding-edge, please see the xorg-edgers PPA.  While Ubuntu does not "officially/formally" support these packages, if you discover problems when installing these debs please feel free to report bugs to launchpad.  However, please make sure to clearly state that you are running packages from this PPA so we know the fixes need to come here.  If you are upgrading from one release to another with this PPA activated, please install the ppa-purge package and use it to downgrade everything in here beforehand. sudo ppa-purge ppa:ubuntu-x-swat/x-updates will do it.  [Directions for packaging drivers can be seen at https://wiki.ubuntu.com/X/DriverBuilding]  More info: https://launchpad.net/~ubuntu-x-swat/+archive/x-updates Press [ENTER] to continue or ctrl-c to cancel adding it   I want to skip the confirmation step  Press [ENTER] to continue or ctrl-c to cancel adding it   How can I automatically confirm the addition of a PPA? Why is potential energy negative when orbiting in a gravitational field? Can Resharper be set to warn if IDisposable not handled correctly? Is there a setting in Resharper 4 (or even Visual Studio itself...) that forces a warning if I forget to wrap code in a using block, or ommit the proper Dispose call in a finally block? Fitting an infinite collection of balls in an infinite dimensional unit ball Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball?  I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if the concept of orthogonality exists, but not every normed linear space has an inner product so it's not possible to apply something like Gram-Schmidt to produce an orthogonal basis. Is there any way around this, or there another approach that can be used? 18.04.01 Not available for server What's the best way of replacing a plug or switch in a house with aluminum wiring? How to convert std::string to LPCWSTR in C++ (Unicode) Indicate How Trustworthy the UpVotes Are Finding conditional distribution Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant).  this is my idea:  $$f_{X \mid X+Y}(x\mid c) = \frac{f_{X,X+Y}(x,c)}{f_{X+Y}(c) } $$  if this is the right definition then i might calculate the righthand side to obtain the distribution but Iam unsure Is the sign of principal components meaningless? Does the sign of the vectors returned by PCA have any significance after I center my variables? "Welcome to emergency mode!" Think it is a fsck problem Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties.