“Prime conspiracy”'s effect on cryptography Recent news reported about the discovery of a "Prime Conspiracy" which can be read about .   In summary, researchers have discovered that the last digit of prime numbers have a greater predictability than random. The cited example is that primes ending in 9, have a 65% chance of the next prime ending in 1. I am curious if any thought has been generated around the impact this might have to cryptography, namely popular encryption methods like RSA?
Do these new insights into prime numbers affect encryption security? reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time?
Doesn't Radagast care about trees? In the book The Two Towers, Treebeard refers to Gandalf as "the only wizard that really cares about trees".  But in Fellowship of the Ring, we met the wizard Radagast the Brown, whose special focus is on animals, especially birds.  Of course, animals aren't trees, but it is difficult to imagine a person loving animals without also caring deeply about trees - especially when that person loves birds most of all, and so many birds are so  dependent upon trees for their survival.  More generally, Radagast seems like an all-around nature lover, and this too suggests that he "really cares about trees".  This leads me to imagine several possibilities:   Treebeard doesn't know Radagast (this seems unlikely - they have both been around for thousands of years, so surely they have bumped into each other at some point)   Treebeard forgot about Radagast, possibly because he was annoyed at Saruman and a bit angry at the time    Radagast actually doesn't care about trees    Is there some other alternative I have overlooked, or is one of the options I mentioned correct?  Or does Tolkien never address this apparent problem? How do I move to a new tab in gedit using the keyboard? If I have multiple tabs open, how do I move from one to another? (Ubuntu 11.10; gedit 3.2.1) Batch Rename Files in Folder Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$. How do Southern people discern Night's Watch deserters from truthful brothers sent south? Night's Watch deserters are sentenced to death, as we've seen in the very first episode of Game of Thrones. A man dressed all in black is apprehended by Stark men and brought before Lord Eddard Stark. Lord Stark sees him as a deserter and sentences him to die (and then executes that death penalty).  However, there are men of the Night's Watch south of the Wall, on a legitimate mission or on some kind of allowed leave. For instance, we have Benjen Stark visiting Winterfell when the king and his party is there. There is also Yoren, who travels around Westeros in search of new recruits.  I could imagine that the Warden of the North (and other lords in the northern part of Westeros) would have a good idea of who would be allowed further south and whose appearance would raise suspicion. But I reckon the southern lords don't have as many visits from the Night's Watch, so they might not know who would be allowed to be there and who wouldn't.  How would lords who are rather unfamiliar with the specific brothers of the Night's Watch discern deserters from brothers sent out for some reason?  That ends my question, but there is a spoiler from the season 5 finale that continues the question a bit more:      Sam is sent to Oldtown to become a Maester, so he can replace Maester Aemon at Castle Black.  But he is riding together with Gilly (a woman!) and her baby (a baby!), which might raise a lot of suspicion from people who don't know what he is sent out to do.  What prevents southern lords of sentencing Sam as a deserter from the Night's Watch? I do as follows  $(1+x)^n= \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+...+\binom{n}{n}x^n$   differentiating in both sides  $n(1+x)^{n-1}= \binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^2...+n\binom{n}{n}x^{n-1}$  then, for $x=1$  $n \cdot2^{n-1} = \sum_{k=1}^{n}k\binom{n}{k}$  From here, I don't know how to proceed, I have tried to multiply both sides for $(1+x)$ to make $n \cdot2^{n}$ but then the summatory changes, I don't know how to prove the equality. Is there a word for someone who won't think critically? Is there a word for someone who won't think through issues and instead comes to others hoping that they will think for them and solve problems for them?  A word that is more specific to a lack of critical thinking skills than "lazy"? Limit of $2^{1/n}$ as $n\to\infty$ is 1 How do I prove that:   $\lim \limits_{n\to \infty}2^{1/n}=1$  Thank you very much. Which part of speech is "Be assured"? Which of the spans in 1.1 is the stronger? Be assured, you are in numerous company if you pick the left one at first glance, where the struts and wires are doing nothing useful at all.   Could you please tell me what is "Be assured" here?  I mean whether it is an adverb, adjective, so on?  And, would you please show me some examples exactly like such the following, considering my original sentence?     Be assured, ... So a buddy and I were playing a game of Magic that turned out to be a really quick game because he got  out on the battlefield early. He was tapping it without attacking because of its obviously awesome ability but I don't know if you can tap a creature for its Inspired (untap-triggered) ability without attacking for it to then be tapped. Are you allowed to tap a creature whenever you want? Setting environment variables on OS X What is the proper way to modify environment variables like PATH in OS&nbsp;X?  I've looked on Google a little bit and found three different files to edit:   /etc/paths ~/.profile ~/.tcshrc   I don't even have some of these files, and I'm pretty sure that .tcshrc is wrong, since OS&nbsp;X uses bash now. Where are these variables, especially PATH, defined?  I'm running  (Leopard). I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses and sheer volume of education into the physical tools and mindset that the other students who took the course had, like Noether's theorem, Lagrangian and Hamiltonian mechanics, statistical methods, and so on.  The courses themselves went well enough. My mathematical experience more or less made up for a lacking physical understanding. However, I still haven't found an elementary explanation of gauge invariance (if there is such a thing). I am aware of some examples, like how the magnetic potential is unique only up to a (time-)constant gradient. I also came across it in linearised general relativity, where there are several different perturbations to the spacetime metric that give the same observable dynamics.  However, to really understand what's going on, I like to have simpler examples. Unfortunately, I haven't been able to find any. I guess, since "gauge invariance" is such a frightening phrase, no one use that word when writing to a high school student.  So, my (very simple) question is: In many high school physics calculations, you measure or calculate time, distance, potential energy, temperature, and other quantities. These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above? Or are these two different concepts? Implementing a sort of "tightened frenchspacing" By default, (La)TeX adds extra space between sentences. The exact rules are in the TeXbook, a summary by TH. can be found . What follows is is my "executive summary":   After any period (or quotation/exclamation mark) that doesn't follow a capital letter, TeX adds a font's "extra space" to the "normal (interword) space". For Computer Modern 10pt, the normal space is 3.33pt and the extra space 1.11pt, resulting in an intersentence space of 4.44pt (+33% compared to the nomal space, assuming no stretch and shrink). A sentence end after a capital letter is indicated by the \@ command: Ann is learning the ABC\@. To denote that a period after a lowercase letter doesn't end a sentence, one may use \@ as well as an explicit space: Prof.\@ Higgins or Prof.\ Higgins.   Intersentence spacing may be brought in line with normal spacing by issuing the \frenchspacing command. (This is the standard setting for certain languages, e.g. the  (n)german option of the babel package.) With \frenchspacing enabled, there's no need to use the \@ command. One may revert to (La)TeX's default behavour by issuing \nonfrenchspacing.  So, there's the choice between extra intersentence spacing and uniform spacing. What is missing in my opinion is a third option I'll call "tightened frenchspacing": normal (interword) spacing between sentences and decreased spacing after non-sentence ending periods. To be somewhat more precise (while still ignoring stretch and shrink), I envision the following:   Adhering to a font's ratio between "normal space" and "normal space plus extra space" (3:4 for Computer Modern), the intersentence space is set to the normal space. The space after non-sentence ending periods equals the normal space multiplied by the above ratio (resulting in a space of 2.5pt for Computer Modern). As with \nonfrenchspacing in effect, the \@ command is used to denote cases of special punctuation.   How should such a "tightened frenchspacing" option be implemented?  EDIT: Thanks to Mico for providing what may well be a stepping stone to a solution. Here's another attempt at clarification:   Simply reducing the spacing after any period is not what I had in mind. There should be a (tangible) difference between end-of sentence spaces and spaces after non-sentence-ending periods (only that the latter should be reduced instead the former augmented). If adhering to a font's normal space/extra space ratio is not possible, a solution that adds a fixed (reduced) space (say, a thin space) after non-sentence-ending periods will be accepted. For compatibility reasons, I'd prefer a solution that works with the traditional use of \@ (i.e allows "tightened frenchspacing" for existing documents without the need to change the document body). Should it turn out that a different set of "special punctuation" macros is needed, I'll accept this as a solution. What is the best method of re-rendering a web page on orientation change? This may be a stupid or trivial question, but I am new to Ubuntu, and have two unsolved problems (flashplayer doesn't play, and scanner doesn't work - in virgin install 13.04), so I started to search in documentation. In the 13.04 specific web site I was browsing the hardware section, and came across something like this: "Open Dash, and select Disk Utilities".  So what is Dash, and where is Disk Utilities?  The application finder could not find any of those. In terminal the "dash" command seems to invoke some sort of a shell. Why does my user number show up in the search box when clicking my profile information? Hey all-  I just noticed that if you click on your username on the top bar, when your profile shows up, in the search bar at the top you'll find an explicit search for your user account.  For my meta account, for example, I see user:157031 whenever I click on my profile.  Is this deliberate?  If so, can I suggest removing that text?  It means that I have to manually delete the contents of the search field if I want to do a search from my profile screen.  Thanks! Combine the first two commits of a Git repository? Suppose you have a history containing the three commits A, B and C:  A-B-C   I would like to combine the two commits A and B to one commit AB:  AB-C   I tried  git rebase -i A   which opens up my editor with the following contents:  pick e97a17b B pick asd314f C   I change this to  squash e97a17b B pick asd314f C   Then Git 1.6.0.4 says:  Cannot 'squash' without a previous commit   Is there a way or is this just impossible? Why does it take so long for a computer monitor to change resolution? Why does it take 2-3 seconds rather than, say, 50 ms?  E.g. my current Windows 7 with strong GPU and 24 inch monitor takes 2-3 seconds. Why so long? I know some might say "it's JUST 2 seconds", but come on... Games produce hundreds of FPS, and monitors work at 60 Hz, so what is it that takes so much time?  I remember this was the same in just about any machine I ever worked with. Calculate which day of the week a date falls in using modular arithmetic In  the main character (he is a mathematician) calculates the day of the week of someone's birthday (19/07/1992 is Sunday). I know (very) basic modular arithmetic but I can't figure out how to do it. Can someone point me to the right direction? It seems fun to do. I'm trying to simulate a rope (that later might get pulled on the ends*) that interacts with some obstacles. So far I created a plane as the floor and a cube as an obstacle, and activated "collision" in the physics menu for both of them.   *(In the end the goal is simulating a .)  For modelling the rope, I tried using a long cylinder with enough subdivisions, and activate "cloth" in the physics menu. This resulted in the cylinder collapsing like a sleeve of a sweater, but I would like the cross section to remain roughly circular. Is there a way to directly avoid this?   Another approach I tried now was just using a a line made up of vertices and edges (with the idea of later adding modifiers to give it a thickness), but this just fell through all the collision objects as it has no faces. Is there good way to do this?