I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on, sure this is true and there's a long proof, but too long to see.
I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on,
What is the easiest/best/most correct way to iterate through the characters of a string in Java? StringTokenizer? Convert the String to a char[] and iterate over that? Something else? $e^{mx}$ in solving second order differential equations In a book I am reading on differential equations, the author writes about the solution to a homogenous, linear, second order differential equation with constant coefficients. The author says something like, "Let us suppose that the solution is of the form $y=e^{mx}$ " . After this the author introduces the characteristic equation of a differential equation of the form mentioned above, and proceeds to describe how to solve it w/ undetermined coefficients, variation of parameters, etc.   How did mathematicians first come up with this "assumption" that the solution was of the form $y=e^{mx}$? And, are there any other forms of solutions for these types of equations? How is it possible that quantum phenomenons (e.g. superposition) are possible when all quantum particles are being constantly observed? I don't understand how quantum mechanics (and therefore also quantum computers) can work  given that while we work with quantum states, particles that this quantum state consist of cannot be observed, which is the most fundamental requirement.  If I am not mistaken, by "observed" we mean interaction with any other particle (photon, gluon, electron or whatever else). So my very important questions:   Aren't the particles this quantum state consists of interacting with each other? Why doesn't that cause the state to collapse? Aren't all particles in the universe interacting with Higgs field and gravitons etc? Why doesn't that cause every quantum state to collapse?   I feel there is something very fundamental in quantum mechanics that I am not aware of, hence I would be very pleased to have these questions answered. What is the lifespan of an SSD drive? Are SSDs comparable to HDDs for reliability and is there published research on this?  Also are all SSD manufacturers at a similar 'reliability level'? Did the Big Bang happen at a point? How to kill all process with given name? What are the differences between a pointer variable and a reference variable in C++? I'm a college student and I just got my first credit card (through Chase). The only reason I got the card is to try and build my credit score. I also have a checking account with Chase, and they let me pay off my credit card any time I want.  For example, right now I have a balance of $92 on my credit card, but my bill won't come until September 2, and the payment isn't actually due until September 27. But through their website I could go ahead and pay it off with money from my checking account.  My question is, is it better for my credit score if I pay off that $92 balance right now, or should I wait for the bill, or does it not matter either way? $3^{n+1}$ divides $2^{3^n}+1$ Describe all positive integers,n such that $3^{n+1}$divides $2^{3^n}+1$. I am little confused about what the question asks-if it asks me to find all such positive integers, or if it asks me to prove that for every positive integer n,$3^{n+1}$ divides $2^{3^n}+1$. Kindly clarify this doubt and if it's the former part, please verify my solution-n=1. Generating hatched filling using Region functionality Which is faster/best? SELECT * or SELECT column1, colum2, column3, etc I've heard that SELECT * is generally bad practice to use when writing SQL commands because it is more efficient to SELECT columns you specifically need.  If I need to SELECT every column in a table, should I use   SELECT * FROM TABLE   or   SELECT column1, colum2, column3, etc. FROM TABLE   Does the efficiency really matter in this case? I'd think SELECT * would be more optimal internally if you really need all of the data, but I'm saying this with no real understanding of database.  I'm curious to know what the best practice is in this case.  UPDATE: I probably should specify that the only situation where I would really want to do a SELECT * is when I'm selecting data from one table where I know all columns will always need to be retrieved, even when new columns are added.   Given the responses I've seen however, this still seems like a bad idea and SELECT * should never be used for a lot more technical reasons that I ever though about. $a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$ So far I have that $a|b$ implies $b=ax$ for some x in the integers and $c|d$ implies that $d=cy$ for some y in the integers.  From here I can see that $gcd(a,c)|gcd(b,d)$ is logically equivalent to $gcd(a,c)|gcd(ax,cy)$ but I am not sure where to go from there. Bogus float inequality due to compiler optimization? How do I make multiple spawn points in minecraft vanilla? Splitting of primes in the compositum of fields If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in all of $L_i$. Does this also hold if the $L_i$ are not Galois extensions of $K$?  The proof that I know regarding the compositum uses the fact that the $L_i/K$ are Galois, so is this true in the more general setting and how would one prove it? Blade of the Ruined King Vs. The Bloodthirster Effects:  +25 attack damage +15% life steal +40% attack speed Passive  Unique: Your attacks deal 5% of the target's current health in physical damage (60 max vs minions and monsters). Active   Unique: Drains target champion, dealing 15% of the champion's maximum Health in physical damage and healing you by the same amount. Additionally you steal 30% of their movement speed for 4 seconds - 60 second cooldown. (500 range)  Cost: 3200g VS.   Effects:  +70 attack damage +12% life steal Passive: Gain +1 attack damage and +0.2% life steal per kill (maximum +30 attack damage and +6% life steal). Half of the bonuses are lost upon death.  Cost: 3200g What is better for what situations, and when should you build BT instead of BotRK and Vice Versa. Which provides more lifesteal, the only variable being the stats of the weapon? I find myself constantly questioning my decision on which weapon to get and would really love some in depth answers. Would be great if any examples given were given from the ADC side of the house. pgfplots: prevent single plot from being listed in legend In order to fill the area between two complex curves I use tables and the vertcat command (see ).   Now I don't want the plotted area to appear in my legend, just the two curves. Is there any command to prevent the yellow area plot from being added to my legend?  this could be a minimal example:  \documentclass{article} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture}   \begin{axis}[area style,axis on top]     \addplot+[mark=none,fill=yellow,draw=none] {0.1*x^2} \closedcycle;     \addplot[mark=none,draw=red,line legend] {0.1*x^2};     \addplot[mark=none,draw=blue,line legend] {0.0*x^2};     \legend{red curve,blue curve};     % \legend{\empty,red curve,blue curve};   \end{axis} \end{tikzpicture} \end{document}   With this legend{...}-command the last plot is skipped, an additional \empty doesn't help, too.  Sadly, drawing the first to plots together with fill=yellow,draw=red is not an option for me. Does anybody know any inside hacks? As far as I know, there is no command shown in the manual. Thanks in advance. Render shadows over a transparent background in cycles I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: I see the following error while trying run the command shown below. I read somewhere that my /boot partition is low on disk space. How can I increase the size of the /boot partition so I can install more software? I have a 500GB hard disk, so there is enough space to play with.  sudo apt-get install libdvdread4  gzip: stdout: No space left on device                                                                                                                                                                                  E: mkinitramfs failure cpio 141 gzip 1                                                                                                                                                                                 update-initramfs: failed for /boot/initrd.img-3.2.0-33-generic with 1.                                                                                                                                                 run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1                                                                                                                                            Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-3.2.0-33-generic.postinst line 1010.                                                                                                        dpkg: error processing linux-image-3.2.0-33-generic (--configure):                                                                                                                                                      subprocess installed post-installation script returned error exit status 2                                                                                                                                            dpkg: dependency problems prevent configuration of linux-image-server:                                                                                                                                                 linux-image-server depends on linux-image-3.2.0-33-generic; however:                                                                                                                                                    Package linux-image-3.2.0-33-generic is not configured yet.    dpkg: error processing linux-image-server (--configure):     dependency problems - leaving unconfigured    dpkg: dependency problems prevent configuration of linux-server:     linux-server depends on linux-image-server (= 3.2.0.33.36); however:      Package linux-image-server is not configured yet.    dpkg: error processing linux-server (--configure):     dependency problems - leaving unconfigured    No apport report written because the error message indicates its a followup error from a   previous failure.                                                                                                           No apport report written because the error message indicates its a followup error from a previous failure.  Errors were encountered while processing:  linux-image-3.2.0-33-generic  linux-image-server  linux-server N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension   Listed below is the output of du  Filesystem              1K-blocks      Used Available Use% Mounted on /dev/mapper/ubuntu-root 712660664 104095912 572363692  16% / udev                      3964792         4   3964788   1% /dev tmpfs                     1591012      1064   1589948   1% /run none                         5120         0      5120   0% /run/lock none                      3977528       684   3976844   1% /run/shm /dev/sda1                  233191    219821       929 100% /boot How can I dynamically invoke a class method in PHP? The class method is not static. It appears that   call_user_func(...)   only works with static functions?  Thanks.