can $a_{n+1}=2a_n+1$ generate prime numbers?
Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
Why are images formed? How do rays of light getting focused at a point create an image?  How are the information of object's shape and dimensions expressed as an image when light rays converge?  Is it understandable by solving wave equation for light? How do I calculate the distance of an object in a photo? How is a second measured? And why is it measured that way? The earth's rotation around itself (one rotation around Earth's own axis) doesn't take exactly 24 hours. It off by some seconds which becomes somewhere around 6 hours per year and 1 day in 4 years(leap year), which brings the question why didn't we modify the measurement of 1 second ever so slightly so as to avoid leap years altogether.   Well how is 1 second measured exactly?  says      the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. []   Well, why is it measured that way? Is there any technical reason like it is easy to measure and can be standardized easily? Or is it really possible to modify the measurement of time? How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something Did this character lose a limb in “The Force Awakens?” Inspired by  which states limb loss as a recurring theme, I started thinking about the final lightsaber battle. After watching The Force Awakens the second time, I was not sure if      Kylo Ren   lost a limb or just lost his lightsaber. We definitely see      Rey slash at Ren’s right hand.   So knowing that context:      Did Kylo Ren actually lose his arm/hand? What effect does serial downvoting have on my rep cap? So I was the victim of serial downvoting the other day, and just had it automagically reversed. Rather than simply removing the previous votes, I was given a bonus lump of rep with the reason "serial downvote reversal". The downvotes and the bonus show up on different days. Since a whole day's quota worth of downvotes equates to quite a fraction of my daily rep cap, what happens in the following circumstances?   I get 200+ rep on a particular day, but I am serial downvoted (later reversed) back to under 200. Does this day count as reaching cap? I get 200+ rep the next day, and I am also credited back the rep. Is the rep bonus immune to the rep cap in the same way bounties are? On the day I get my rep bonus, it is enough to take me to 200 rep. Does this count as capping? Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. Image viewer without large window borders I am looking for an image viewer software without very large window borders. I don't need any type of toolbar, just a simple image viewer.   It would also be fine if there is no top bar to close, minimize or maximize window. What is the Fastest a Character Can Move in One Turn? At level 20 with any magic items in the DMG, and any spells, what is the fastest a DnD 5e character could move using their movement?  (i.e., not including teleportation) Start script on open blender file Book series where they collect either stones or gems I'm looking for a fantasy book I read about 10 years ago. I don't remember a lot of it, but there are a few things I still know:   The setting was in medieval times. There was no traveling dimensions. I can't recall whether there was magic or not. It wasn't on Earth, but on some invented world. I believe it was a trilogy, with a focus on the son of some nobility. The goal for the characters was to collect a certain number of stones or gems (I believe there were 5 of them), all in certain colours. Every stone has a certain power attached to it. I believe it started out with one or two characters, but the group grew and every character had a certain colour stone at the end. What is important about the gems is that they are unique. It's not part of a bigger magic system, but they are artefacts from somewhere.   There are some minor incidents I can recall vividly:   They build up a fellowship with some stereotypical fantasy builds: an old school fighter, a minstrel, a female archer and probably more. At one point, they mention that if one of the guards in a country falls asleep on duty, he will be kept awake by his garrison/brothers in arms for seven days. At some point the minstrel falls asleep, and the old fighter teaches him how to stay awake. They travel through a forest full of leeches and other insects to reach one of the stones. Also, I seem to recall there was a member of an underground evil organisation among the nobility, who tried to poison someone and were able to do that because they built up an immunity to the poison.   Does anyone has a clue what this series might be?   Suggestions by kind people, but not what I'm looking for, are:   The Seventh Tower Deltora Quest Diadem Westria Sword of Heavens series What happens when anti-matter falls into a black hole? Let's say a black hole of mass $M$ and a very compact lump of anti-matter () also of mass $M$ are traveling toward each other. What does an outside observer see when they meet?  Will they blow themselves apart in a  matter/anti-matter reaction? Or will their masses combine, never quite meeting in the infinite time dilation at the event horizon? Implementing the decorator pattern in Python Can Minecraft Pocket Edition play with a desktop Minecraft? Some people say that mass increases with speed, some people say that the mass of an object is independent of its speed.  I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be 'true' is any sense of the word.  Either mass increases or it doesn't, right?  Can't we just measure it, and find out which 'interpretation' is right? e.g.: (in some sophisticated way) heat up some particles in a box and measure their weight?  ------------------------------  UPDATE  Right, so I've got two identical containers, each with identical amounts of water, each on identical weighing scales, and each in the same g field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and g is constant, does this mean that the m in w=mg has got bigger? Pgfplots: Linewidth when using \ref in custom legend Word with the same meaning as "good moments" or "pleasant memories"? Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. How find the maximum value of $|bc|$ Question:     Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$   It is said this is answer is $$|bc|\le \dfrac{3\sqrt{3}}{16}$$  My idea: let $z=1$,then $$|a+b+c|\le 1$$ let $z=-1$,then $$|a-b+c|\le 1$$ let $z=0$, then $$|c|\le 1$$  let $|\theta|=1$,then we have $$|a(z/\theta)^2+b(z/\theta)+c|\le 1\Longrightarrow |az^2+b\theta z+c\theta^2|\le 1$$ and only this can't solve this problem,Thank you I guess I've been somewhat ignorant when it comes to the finer details of pcb layout. Lately I've read a couple of books that try their best to lead me on the straight and narrow. Here is a couple of examples of a recent board of mine, and I have highlighted three of the decoupling caps. The MCU is a LQFP100 package and the caps are 100nF in 0402 packages. The vias connect to ground and power plane.     The top cap (C19) is placed according to best practices (as I understand them). The other two are not. I haven't noticed any problems. But then again the board has never been outside the lab.  I guess my question is: How big a deal is this? As long as the tracks are short, does it matter?  The Vref pins (reference voltage for the ADC) also have a 100nF cap across them. Vref+ comes from an onboard TL431 shunt regulator. Vref- goes to ground. Do they require special treatment like shielding or local ground?    EDIT    Thanks for great suggestions! My approach has always been to rely on an unbroken ground plane. A ground plane will have the lowest possible impedance, but this approach may be too simplistic for higher frequency signals. I've made a quick stab at adding local ground and local power under the MCU (The part is an NXP LPC1768 running at 100MHz). The yellow bits are the decoupling caps. I'll look into paralleling caps. The local ground and power are connected to the GND layer and the 3V3 layer where indicated.  The local ground and power are made with polygons (pour). It's going to be a major rerouting job to minimize the length of the "tracks". This technique will limit how many signal tracks can be routed under and across the package.  Is this an acceptable approach? How do I remap the keys on my keyboard? I broke my laptop and had to buy a new one with a new keyboard that I am struggling to get familiar with, especially with the keys like home, page-up, page-down and end, which aren't standardized at all.  I found myself doing this for like the 5th time in the last several years, and I think it is really annoying.   Is there an app that can reliably map one key to another?  By reliably I mean that key mapping works with Shift, Alt, and other modifiers.  UPDATE  Windows 7, Samsung Chronos 7