I have been trying to share information concerning a conjecture that for all Prime numbers greater than 3, the difference between the their squares, modulus 24, is zero. I have tested this for the first 6 million primes without a contrary example. I am continuing the search but this is extremely time consuming. I realise that this information is trivial and only worthy of the briefest of notes, and I cannot find where to pass on the information. Does anyone know  where or who I can ask whether this is a known conjecture?
Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$.   This is what I got so far. I figured that since $p,q$ are bigger than $5$, there are only odd primes for this conjecture. Then I can rename $p=2m+1,q=2n+1,m,n∈Z$.  We want to show $(p^2-q^2 )=24k,k∈Z$. By substituting $2m+1$ for $p$ and $2n+1$ for $q$, we get  $((2m+1)^2-(2n+1)^2)$. If we expand it, we get $4(m^2+n^2+m-n)$ Q. I think I can make $4(m^2+n^2+m-n)$ look like $24k$. How should I approach ?  And my prof. commented that "I would take this proof in small steps.  Rather than trying initially to prove that $24 | p^2-q^2$, look at the progress you already have.  You proved $4 | p^2-q^2$.  Can you prove 8 is a factor?  What about 3?  Then see if you can put together what you have to get 24 as a factor."  I am stuck here. Any helps?
I used following command for port scanning of my machine  nc -zv 192.168.1.1 1-100  but I want to filter only succeeded message from  following output.Hence i used the following command   nc -zv 192.168.1.1 1-100|grep succeeded   But no use, still it shows full output   nc: connect to 192.168.1.1 port 1 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 2 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 3 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 4 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 5 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 6 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 7 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 8 (tcp) failed: Connection refused nc: connect to 192.168.1.1 port 9 (tcp) failed: Connection refused How to write a while loop with the C preprocessor? Is there an ideal PWM frequency for DC brush motors? 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Like in here:  There is still charge on one plate of the capacitor, which will try to go to the other plate to make $\Delta V = 0$ across the capacitor, so there has to be a current $I$.  But as more charge goes to the other side of the capacitor, the more repulsion these will cause to more coming towards them. So I am assuming current cannot just be constant?  I am starting from $$ I = C \frac{dV}{dt}, $$ but I know neither $V(t)$ nor $I$. Is RSA key size the size of private key exponent? How to identify polygons with "flag" sliver errors Working in ArcMap, I have come across errors in a polygon layer that I will call "flag slivers" (taken from similar language in parcels call "flag lots").  These sliver polygons (see image below) are typically composed of just one additional node that user has  accidentally added.    In the images below, the "flag" and the "flagpole" are one single-part feature where the "flagpole" just overlaps itself.  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To make a JavaScript class with a public method I'd do something like:  function Restaurant() {}  Restaurant.prototype.buy_food = function(){    // something here }  Restaurant.prototype.use_restroom = function(){    // something here }   That way users of my class can:  var restaurant = new Restaurant(); restaurant.buy_food(); restaurant.use_restroom();   How do I create a private method that can be called by the buy_food and use_restroom methods but not externally by users of the class?  In other words, I want my method implementation to be able to do:  Restaurant.prototype.use_restroom = function() {    this.private_stuff(); }   But this shouldn't work:  var r = new Restaurant(); r.private_stuff();   How do I define private_stuff as a private method so both of these hold true?  I've read  a few times but it doesn't seem like "private" methods can be called by public methods and "privileged" methods can be called externally. What constitutes protons? When I see pictures, I can't understand. Protons are made of quarks, but some say that they are made of 99% empty space. Also, in this illustration from Wikipedia, what's between the quarks? I built a simple bridge rectifier.         &ndash; Schematic created using   I connected the output of the function generator with CH1 of my oscilloscope and  with the input of the rectifier (using a T-Piece). The output goes into CH2.   The original wave looks like this:    After connecting CH2 the picture looks like this:   While CH2 looks good,  CH1 now shows the wave I would expect without capacitator but not the expected original wave.   Could somebody explain why? I'm using a Gwinstek GDS-1072A-U Oscilloscope and a Gwinstek SFG-1003 function generator. In Avengers: Endgame,       Thanos was able to stop and catch Mjolnir and Stormbreaker mid-air as both Captain America and Thor used/called them BEFORE he got the Infinity Stones.   Since this character is clearly not worthy and had no super powers at the time, how were they able to wield Thor's Mjolnir and Stormbreaker, when even the Hulk couldn't do it in The Avengers? Valid characters in a Java class name What characters are valid in a Java class name? What other rules govern Java class names (for instance, Java class names cannot begin with a number)? Powershell Parameters Why does the /boot partition get updated when I install software Based on some recommendations I read online, on install of a new (Ubuntu) system, I made my /boot partition separate from / and small in size (only around 50 MB). It makes sense to me that you would only need limited files in that partition, and the boot sequence only needs so much data initial data on boot so no need (or so goes my thinking) to make it much larger.   I was surprised as I started using my system and installing software was failing due to the following (or similar) issue:  gzip: stdout: No space left on device cpio: write error: Broken pipe E: mkinitramfs failure cpio 1 gzip 1 update-initramfs: failed for /boot/initrd.img-3.2.0-18-generic with 1.   I have already seen  and am planning on doing a reinstall with a larger /boot partition. My questions are:  Why does the /boot partition data get updated when I am only installing software?  obviously 50 MB is too small for my needs, so what would be a reasonable /boot partition size?  Does it even make sense to have separate boot partition? after all it could simply use all the disk space it wants in the / partition. We usually say that the ultimate source of energy for earth is the Sun. So this means that sun creates energy but according to the law of the conservation of energy energy can neither be created nor destroyed, so how does the energy come into the sun? That is just an example. We know that energy is transformed from one form to another but is not created and destroyed. But that is confusing because there must be an ultimate source of energy. How is  energy created in this source? If it is related to the atomic level, then how can atoms possess energy? Is the law of the conservation of energy untrue? I have a scenario. (Windows Forms, C#, .NET)   There is a main form which hosts some user control. The user control does some heavy data operation, such that if I directly call the UserControl_Load method the UI become nonresponsive for the duration for load method execution. To overcome this I load data on different thread (trying to change existing code as little as I can)   I used a background worker thread which will be loading the data and when done will notify the application that it has done its work.   Now came a real problem. All the UI (main form and its child usercontrols) was created on the primary main thread. In the LOAD method of the usercontrol I'm fetching data based on the values of some control (like textbox) on userControl.    The pseudocode would look like this:  CODE 1  UserContrl1_LoadDataMethod() {     if (textbox1.text == "MyName") // This gives exception     {         //Load data corresponding to "MyName".         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage.     } }   The Exception it gave was     Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on.   To know more about this I did some googling and a suggestion came up like using the following code  CODE 2  UserContrl1_LoadDataMethod() {     if (InvokeRequired) // Line #1     {         this.Invoke(new MethodInvoker(UserContrl1_LoadDataMethod));         return;     }      if (textbox1.text == "MyName") // Now it wont give an exception     {     //Load data correspondin to "MyName"         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage     } }   BUT BUT BUT... it seems I'm back to square one. The Application again  become nonresponsive. It seems to be due to the execution of line #1 if condition. The loading task is again done by the parent thread and not the third that I spawned.  I don't know whether I perceived this right or wrong. I'm new to threading.  How do I resolve this and also what is the effect of execution of Line#1 if block?  The situation is this: I want to load data into a global variable based on the value of a control. I don't want to change the value of a control from the child thread. I'm not going to do it ever from a child thread.   So only accessing the value so that the corresponding data can be fetched from the database. Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$ Show that for a non-negative random variable $X$,   $$\mathbb E(X)=\int_0^\infty \mathbb P(X\ge x) \, dx.$$   I started with  $$\mathbb E(X)=\int_0^\infty x \, dF(x)=\int_0^\infty \int_0^x dt\,dF(x).$$ This is my try. How can I add a newline in the text of a label in WPF such as the following?  &lt;Label&gt;Lorem    ipsum&lt;/Label&gt;