How can I show this? I was doing some proves, and this question came out of nowhere, but I felt curious about this and so I tried this. The question is that given any $k \in {{\Bbb Z}^ + }$ such that $k \geqslant 5$, show that ${6^k} + 1$ is composite. This question looks and feels extremely difficult to prove for me, and so I want help from this. Luckily, I did some proves such as using By Induction, By Contradiction, and more, and nothing worked. Also, I tried looking this problem through the Internet, and nothing... What I tried to do was the following, though: Show that if $$A = \{ {6^k} + 1|{6^k} + 1{\text{ is prime}}\} $$then $A = \{ 7,37,1297\} $. Especially, only when $k = 1$, $k = 2$ or $k = 4$, ${6^k} + 1$ is prime. Or equavilently, ${6^k} + 1$ is prime iff $k = 1$, $k = 2$ or $k = 4$. Any ideas? Thanks, Joshua. Update: This question is already answered. Thanks, everyone for helping. Update 2: I think that I can prove this (but not completely). Proof: Let $k \in {{\Bbb Z}^ + }$ and odd. Then $k = 2m + 1,m \geqslant 2$. $\Rightarrow {6^k} + 1 = {6^{2m + 1}} + 1 = 6 \cdot {36^m} + 1$. Since $36 \equiv 1(\bmod 7) \Rightarrow 6 \cdot {36^m} + 1 \equiv 6 \cdot {1^m} + 1 = 6 \cdot 1 + 1 = 7 \equiv 0(\bmod 7)$. Hence when k is odd, $6^k + 1$ is composite. If $k = ab$ and $b$ is odd, then $6^k + 1 = 6^{ab} + 1 = 6^{ab} + 1 \Rightarrow 6^a + 1|(6^{ab}+1)$.
Does anyone know whether there are only finitely many of primes of the form $6^{2n}+1$, where $n$ zero or any natural number?
I am trying to figure out how to make the space around the sun glow and than fade away into darkness.  The sun was made in blender using cycles render and several noise textures. The glow I have around my sun was made using Blenders Gurus tutorial for earth, atmosphere effect, to create a red glow. The sun will be animated so using composition is not going to work for me (sadly);  I realized one of the reasons why the method is not working is I am using surface rather than a volume shader, but As I tried to do it using volume shaders, I could not figure out what direction to take. My linear voltage regulator is overheating very fast I'm using a 5 V / 2 A voltage regulator () without a heatsink. I'm testing the circuit with a microcontroller (PIC18FXXXX), a few LEDs and a 1 mA piezzo buzzer. The input voltage is aprox. 24 VDC. After running for a minute, the voltage regulator starts to overheat, meaning it burns my finger if I keep it there for more than a second. Within a few minutes it starts to smell like it's burnt. Is this a normal behavior for this regulator? What could cause it to heat that much?    Other components used in this circuit:  L1:   R2:   F1: How can I play encrypted DVD movies? My machine can't play encrypted DVDs on a fresh install.  How do I add this capability?  Another useful bit of information would be what programs are best for playing DVDs, once I'm able to do so.  See the .  Will I be able to play DVD movies from ? How to create a squiggle arrow with some text on it in TikZ? I need to define a new command to create a squiggle arrow with some text on it. Something similar to what \xrightarrow command produces but with wiggly arrows as in \rightsquigarrow. The length of the arrow should automatically be adjusted to fit the text above it and I do not know how to handle this part in TikZ. Any ideas would be much appreciated. Using image sequence of medical scans as volume data in cycles I know it's possible to do this in BI, but I need your help to do this in cycles. For now, I have a setup to use an image sequence as volume but it only shows 1 slice/frame. I'm using generated texture coordinates for easy mapping (0-1 on each axis). The images are mapped on XY and I would like to use normalized frame# as Z coordinate to map the sequence as a volume.       Images sequence can be found  (I'm using 16-bit TIF files of MRBrain in this exemple).    Any solution is welcome but try to keep it simple and user friendly :).  Final Edit: I had a hard time choosing between Lemon's and PGmath's answers. Lemon's is more user friendly and flexible, and uses the image sequence without any change inside blender. PGmath's uses a 3rd party converter to make the sequence 1 single image, but allows GPU rendering. Well, ImageMagick's command line is : "convert +append *.tif anim.tif" from the folder where the sequence is unzipped. Not so hard. That's why I accepted PGmath's answer.      Now, i played with it, and here's the result.  The "volume" object contains the volume shader (playing with the color ramp gives funny things) and the boolean modifiers (turn them off to see the complete volume, but reduce the final density multiplier for better view). The "Slice Control" arrow empty controls the slice position/rotation. The "Texture Mapping" object controls...the mapping, if it need some tunning (position, rotation, scale, and then tiling if necessary?). Comma after introductory words, phrases, clauses: unacceptable, obligatory or optional? I am no native speaker and always confused about the comma in introductory phrases, in particular in prepositional phrases. Is there any hard rule when a comma must be set? If I make a google search for certain phrases, I often find both variants. Typical examples where I am not sure whether a comma must be set are:        First[ly][,] I must phone mother   In this case[,] we must...   For simple problems[,] the algorithm...   From Lemma 1.2[,] we obtain...   For a typical user[,] the algorithm...   In our theory[,] we...   A dark line on the horizon[,] the mountain range we were headed for seemed somehow not as threatening as when we had started out on the trip.   Avidly reading her novel[,] Jane did not notice that her stop was approaching.   To be sure[,] it is not easy to see a solution.   When you get to Rome[,] give me a ring. I want to start using the terminal more often, but I don't know what are the different commands available to me. Is there a way to list all the different commands that I can make use of? How to debug an unmet dependencies problem? in -> inner  out -> outer / (outter?)  What is the history or set of rules behind why 'inner' doubles the 'n' but 'outer' doesn't double the 't'? How can momentum but not energy be conserved in an inelastic collision? How can I control a servo motor with the PWM output of a microcontroller? I wanted to control the direction of servo motor with a switch. I am using an Atmega32.  Outputs are PIND4 and PIND5 and the input is PINA3, but the two servo motors turn only in 1 direction.  #ifndef F_CPU  #define F_CPU 1000000UL   #endif  #include &lt;avr/io.h&gt; #include &lt;util/delay.h&gt; #include &lt;avr/interrupt.h&gt;   int main(void) { DDRD= 0xFF ; DDRA=0x00; PORTA =0x00; // FAST PWM TCCR1A |= 1&lt;&lt;COM1A0 | 1&lt;&lt;COM1A1 | 1&lt;&lt;WGM11 |1&lt;&lt;COM1B0|1&lt;&lt;COM1B1 ; TCCR1B= 1&lt;&lt;WGM12|1&lt;&lt;WGM13 ; ICR1 = 19999 ;  TCCR1B |=1&lt;&lt;CS10 ; while (1) {    if(PORTA==0x08) //if pina3 is 1    {     OCR1A = ICR1 -2000 ;   //turn servo 1     _delay_ms(1000);          OCR1B=ICR1-2000;    //turn servo 2     _delay_ms(1000);     }     if(PORTA==0x00)// if pina3 is 0     {       OCR1A = ICR1 -900 ; //turn servo 1 in the other direction       _delay_ms(1000);               OCR1B=ICR1-900;     //turn servo 2 in the other direction       _delay_ms(1000);      } }       return 0 ;  } How to find out which (not installed) Debian package a file belongs to? On Debian systems (and derivatives):  $ dpkg --search /bin/ls coreutils: /bin/ls   That is, the file /bin/ls belongs to the Debian package named coreutils.  But this only works if the package is installed. What if it's not? Accepting an answer should automatically upvote it as well Looking for a curly script capital A A math book I'm studying uses a symbol and I'd like to be able to reproduce it.  I used detexify to search for it, but was unsuccessful. Most likely it is the letter 'a' in some font I'm not familiar with.   Here's a zoomed in image of the symbol:    Here it is in context: How do I move child windows without moving or minimizing parent in GNOME 3? I have Ubuntu 17.10 with GNOME 3.  What happens: I have (for example) LibreOffice Calc open, with some data columns in it.  I highlight the columns I want to chart, then I select INSERT > CHART.  The chart child window dialog pops up on top of my data, so I want to move it to the side.  When I grab the top bar and move it, the parent (spreadsheet) window minimizes (I think it actually disappears).  Continuing the above example, if I click FINISH on the chart window, I can then see the minimized spreadsheet application, which is the window title bar plus the menu bar just below it.  This seems like really odd behaviour and certainly falls under the category of "not being useful to the user", since the whole idea of moving the child window in the first place was just to be able to better see the spreadsheet!  In other applications, for example the PDF viewer, a slightly different problem occurs.  When I try to move the child window (for example the print dialog) the parent does not minimize but instead follows the child window around.  This is less surprising than having the parent disappear entirely but equally useless in terms of allowing the user to see the information present in the parent window.  Update 2017/11/06  Pomsky's answer below fixes the problem, but leaves me wondering why the window manager is making the decision that all child windows have modal behaviour (or don't), and not the applications themselves; and further, why modal behaviour prevents moving the child screen around when it obscures useful info on the parent screen.  Neither of these seem like good design to me... How do you interpret a significant but weak correlation? Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, suppose that $\mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$. The following questions about eigenvalues is then natural and I was thinking it would be an application of Cayley-Hamilton but I am having trouble writing out a proof.  How do we show that $M$ and $N$ have the same eigenvalues?  Added (because this question is now target of many duplicates, it should state its hypotheses properly). Assume that all the mentioned values of $k$ are nonzero in the field considered; in other words either the field is of characteristic $0$, or else its prime characteristic $p$ satisfies $p&gt;n$. I would like to look at my referral traffic over time on a per domain-basis in Google Analytics. This is what I have done:   Set New visits as advanced segment  Navigated to traffic-sources -> Referrals Checked a coulple of domains Clicked plot rows   The individual domains now show up in the graph, but the problem is that "all traffic" is also visible. All traffic is much bigger than the graphs for individual domains. This makes it very hard to see any changes as they look all values are Zero.  Is there any way to hide "Visists" from the graph in order to zoom in on the vertical axis? Light clock with mirrors on a moving train Area 51 login no longer redirects nor logs in