How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime? 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.
How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime? 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,
How to place the intercept of x and y axes at (0 , 0) and extend the x and y axes to the edge of the plot Jordan normal form and invertible matrix of generalized eigenvectors proof Struggling to find a place to start with this proof- just began learning about Jordan normal.  Given a 2-by-2 matrix $A$ and a Jordan normal form matrix $J_{\lambda}$, there exists a matrix $S = [v1, v2]$ s.t. $$S^{-1}AS = J_{\lambda}$$ That is, $J_{\lambda}$ is similar to $A$.  WTS: $$(A - \lambda I)v_{1} = 0 $$ $$(A - \lambda I)v_{2} = v_{1}$$    I was going to try to start with the similarity condition and work into producing the two statements below.  Transform the similarity condition to: $$(A = SJ_{\lambda}S^{-1}) * v_{1}$$  Rewrite our first statement to prove as: $$Av_{1} = \lambda I v_{1}$$   Substitute: $$\lambda I v_{1} = SJ_{\lambda}S^{-1}v_{1}$$  But I've been continually getting stuck.  Tried a few ways other than this.  What intuition am I missing? In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show this?  Ideas: The sum of the diagonal entries of $M$ is the trace, which is also the sum of the eigenvalues. The product of the eigenvalues is the determinant. All the diagonal entries of $A$ must be real and strictly positive. Also we have $|m_{ij}|\leq \sqrt{m_{ii}m_{jj}}\leq \frac{m_{ii}+m_{jj}}{2}$. So far nothing is working...  We might try to write the Gram matrix $m_{ij} = \langle v_i, v_j \rangle$, and then $\prod_i m_{ii} = \prod_i \|v_i\|^2$. Now we'd need to show that the determinant, which is a polynomial in the entries of $v_i$, is less than $\prod_i \|v_i\|^2$ somehow. A lot of my scripts stopped working after installing Ubuntu 15.10 and I have trouble remembering this new name as I was used to ethx and wlanx for years. Is there a reason for this change? Do I have to get used to this or can I just rename it and go back to good old eth0?  $ ifconfig  enp0s25   Link encap:Ethernet  HWaddr 77:5a:5e:a6:86:d5         inet addr:192.168.31.239  Bcast:192.168.31.255  Mask:255.255.255.0       inet6 addr: fe80::725a:b6ff:fea6:86d7/64 Scope:Link       UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1       RX packets:4833 errors:1332 dropped:0 overruns:0 frame:666       TX packets:5589 errors:0 dropped:0 overruns:0 carrier:0       collisions:0 txqueuelen:1000        RX bytes:3266446 (3.2 MB)  TX bytes:1046654 (1.0 MB)       Interrupt:20 Memory:d3500000-d3520000 Question  I want to write a book in Arabic (I have an Arabic keyboard, so I don't need a converter). By Arabic book I mean literally Arabic:    title, table of contents  etc    all in Arabic.   The equations, however, I want to write them in English/Latin.  Related    This link helped a lot but it was for Arabic texts in an English main document.    I already checked these two (and plenty other sites). Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! How to capture all std outputs K out of N encryption Is there any way to recover deleted files which may have since been overwritten? I think it would be nice to display tags in alphabetical order. Currently, tags are displayed in order of creation.  As a workaround, I remove, and then add the tags, which is a bit dumb, but I like 'em in alphabetical order.  It's probably pretty trivial to implement, and nice to have! Fibonacci sequence and Euclidean algorithm's connection. Let $f_1, f_2,\dots,f_i,\dots$ be the Fibonacci sequence with $f_1=f_2=1$. For any $a,b\in\mathbb{N}$ with $a&lt;b$, suppose the Euclidean algorithm takes $n$ long divisions to compute $\gcd(a,b)$. If $a\le f_j$, then $n\le j-1$, for $j\ge2$.   I want to prove this problem. But I can't see any connection between Fibonacci and Euclidean algorithm. If anyone can give a hint about first step, it will be very thankful.  My guess is to use induction. Often while playing Pathfinder, my group will run into problems where rules for 3-dimensional play simply don't exist, so far as we can find. Arguments will come up over things like carrying-capacity, movement speed, area-effects, and a variety of other things.  Simply put, are there rules for 3D play for Pathfinder, and if so where can I find them? Is there any way to specify a suggested filename when using data: URI? If for example you follow the link:  data:application/octet-stream;base64,SGVsbG8=  The browser will prompt you to download a file consisting of the data held as base64 in the hyperlink itself.  Is there any way of suggesting a default name in the markup?  If not, is there a JavaScript solution? I am pretty sure that most of us here, or at least those do graphic design for living, have found themselves in following situation:   You agree with a client to do the design, agree on price, schedules, formats etc… You work hard, make adjustments, collaborate with client; everything goes well in the beginning. At some point you realise that the client does not listen your professional advice regarding design and prefers his/her own ideas that might not look so professional (choice of fonts, colors, odd choices of elements, shapes etc…). No matter how hard you try, the client is persistent and will not accept the designer’s advice. You get back to work, try to make the best out of it, and the final result is not so good, mainly because you had to stick to the client’s wishes. Work is shipped, payment done, and you don’t want to include this work in your portfolio because it is not good. After all, it is not your work, it is something that was physically created by you as a designer, but the ideas, concept, story, and message came out as a result of nonprofessional person, your client.   Ok, they pay for it, and as long as they are happy this is fine. But, the more compromises you make, the more time you spend working on bad design. If you are a young designer on a quest to get recognition, you don’t want to work just for the sake of money and then face the lack of good design in your portfolio.   Maybe at the beginning of your career you don’t care that much, but later if life this might be a big issue. At some point you will be approached by a big client that might offer you long-term contract, but first they will want to see your work. No point explaining to them what you did and why you made so many compromises.   They will not be interested in that.  Every project is precious as your time and you want to do use it wisely and produce great work.   How would you deal with such a situation? Thermodynamics of supercooled water Could 'cogito ergo sum' possibly be false? iOS crash log catch, debug info.. Catch and send via email to the Dev team Searching a virtual printer, that sends emails I need a software, that emulates a printer, converts the printed documents to PDF/TIFF and send it via eMail to a specified address (for windows).  The virtual printer / PDF conversion is possible with a lot of PDF-converters. But I need a software, which is also able to directly mail me these PDFs. It's always the same mailaddress, and I have an SMTP server.  Any suggestions? Etymology of "history" and why the "hi-" prefix? According to , history comes from the same root as story. If they are from the same word, where does hi- come from? Is it just because of the English habit of taking names from other languages verbatim or is there something more to it? Using Either a Comma or a Semicolon My teacher was explaining how sometimes a comma isn't enough to make a sentence correct, and that sometimes you have to use a semicolon. I am having trouble figuring out when you should use which.   Can someone please explain?