Problems in solving Number Theory problems (specifically Congruences) I am very new to Number Theory, I understood the basic concepts and theorems but what I am struggling at is that I am not able to apply them on questions, and basically I am not able to process the problem in 'number theory' theorems rather I am thinking from an absolute manner. I have solved around 15-20 divisibility problems of Number Theory and I still have no idea how you deal with congruences in a real problem, for example: Prove that for positive integer $ n$ we have $169| 3^{3n+3}-26n-27$. My approach to this problem was by simplifying the expression as much as I can but it involved congruences, which didn't even strike me. Even when I am reading congruence questions' answers, I am not able to simply understand Now I have a number of questions:  first, does it take time to develop that 'intuition' while solving problems? second, can elementary number theory be mastered through sheer practice? third, is there a good resource to study and practice congruences?
Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$ $ 169$ | $3^{3n+3}-26n-27$ ?  Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that  $ 169x= 3^{3n+6}-26n-27-26$ $ 169x= 27*3^{3n+3}-26n-27-26$ $ 169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$  An integer $m$ exists so that   $ 169x= 26*3^{3n+3}+169m-26$ ($ 13x= 2*(3^{3n+3}-1)+13m$)  Now I'm stuck although it looks simple. Thanks for any input in advance.
Is it possible to generate electric energy from the radiation of radioactive materials I wonder if it is possible to generate electric energy from the radiation of radioactive materials like nuclear waste? If it is then wouldn't that also mean that it could be used as an energy source for a very long time since the radiation takes a long time to decay? Short story (+radio play?) in which astronauts outside their ship slowly drift apart in space Looking for a short story in which the crew of a spaceship for some reason find themselves outside the ship, drifting apart in outer space. They can communicate via their radios but are individually drifting apart.  One by one, they lose communication as they drift further away from each other.  Might have also been adapted into a radio play? Why is carbon dating limit only 40,000 years? Is there any way to tinker with the iPhone SDK on a Windows machine? Are there plans for an iPhone SDK version for Windows?  The only other way I can think of doing this is to run a Mac VM image on a VMWare server running on Windows, although I'm not too sure how legal this is. Can someone help with an explanation and some references, in order to understand the nature of return pulses within a LiDAR system. What is a referee in the manuscript submission process? Please submit, with the manuscript, the names, addresses (including   countries), and e-mail addresses of five potential referees who are   outside the authors' institutions and must not have conflict of   interest with the authors.   Do you think I should name 5 researchers from a similar area that are not affiliated with my university and co-authors? Is this what referees are? proof that topologist sine curve is not locally connected I know that this question has been asked before, but it is still not clear to me why the topologist sine curve is not locally connected.     The topologist's sine curve: Let $S$ denote the following subset of the plane. $$S = \{ (x, \sin(1/x)) \mid 0 &lt; x \le 1\}.$$ The set $\bar{S}$ is called the topologist's sine curve, which equals the union of $S$ and the vertical interval $0 \times [-1,1]$.   My attempt was:  Fix a point $p = (0, t)$ where $t &gt; 0$, and consider a neighborhood $N = B(p,\delta)\cap\bar S$, where $B(p,\delta)$ denotes a ball centered at $p$ with radius $\delta$.  Then, I want to show that every open subset of $N$ is disconnected, so there can't be connected neighborhood of $p$ contained in $N$.  But, I am having a trouble with establishing that every open subset of $N$ is disconnected.  How can I prove the claim? Got NODATA issue: 'NODATA' (does the network require authentication?) I'm looking to get data such as Size/Capacity, Serial No, Model No, Heads Sectors, Manufacturer and possibly SMART data. Which is correct: "This is her" or "This is she"? Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"? mkdir -p for files Prove that if $a \mid b$ then $\gcd(a,b) = |a|$ I couldn't find much on the new Lucene-based search, and don't know much about the engine in general, so I'm asking this question.  Does anybody have a description of how search results are ranked?  Does the rank calculation include:   rep of post's author? views? unique referers? PageRank of referers? rank of questions that link to the post? whether a question is closed? vote count? answer count? Can prematurely powering down a computer cause bad sectors in a hard drive? How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss? If you toss a coin $n$ times, there are $2^n$ possible sequences of heads and tails. Let $E_n$ be the set of sequences which do NOT contain two consecutive heads and $e_n$, the number of sequences in $E_n$.  Thus $E_3$ $=$ $\{$ $TTT$, $TTH$, $THT$, $HTT$, $HTH$ $\}$ and $e_3$ $=$ $3$.  How many elements of $E_n$ have $T$ as the first toss?  Please can anyone help me out here? How can I keep low resolution textures pixelated? Destructors of builtin types (int, char etc..) In C++ the following code gives a compiler error:  void destruct1 (int * item) {   item-&gt;~int(); }   This code is nearly the same, I just typedef the int to another type and something magic happens:  typedef int myint;  void destruct2 (myint * item) {   item-&gt;~myint(); }   Why does the second code work? Does an int get a destructor just because it has been typedefed?  In case you wonder why one ever would like to do this: This comes from refactoring C++ code. We're removing the standard heap and replacing it with selfmade pools. This requires us to call placement-new and the destructors. I know that calling destructors for primitive types is useless, but we want them in the code nevertheless in case we later replace PODs with real classes.  Finding out that naked int's don't work but typedefed ones do was quite a surprise.  Btw - I have a solution that involves template-functions. We just typedef inside the template and everything is fine. How to Translate Content type name using i18n module? I am developing a multilingual site that supports English and Norsk bokmal. I have set English as my default language. I wanted to translate the contents and the other things in the site.  My real problem started when I found that I was unable to see the translation of the content type's name. I went to admin/config/regional/translate/translate and translated everything, but still my content type's name is not translated.  I have attached a screenshot of the translation page. Still I don't find a way to translate the content type's name. How to list the properties of a JavaScript object? Say I create an object thus:  var myObject =         {"ircEvent": "PRIVMSG", "method": "newURI", "regex": "^http://.*"};   What is the best way to retrieve a list of the property names? i.e. I would like to end up with some variable 'keys' such that:  keys == ["ircEvent", "method", "regex"] Run cmd.exe in Administrator mode without explorer.exe I've recently had trouble with explorer.exe not working.  I found a possible solution , using .  The problem is that I cannot "right-click Command Prompt, and then click Run as administrator", since explorer.exe does not work for even a second without crashing.  I tried to use the  to open cmd as administrator, but I get an error 1327 because I have a blank password.    My question is, is there any other way to run cmd.exe with admin privileges?  If not, is there any way to run sfc.exe without admin privileges?  If not, does anyone have solution to my original problem of explorer.exe stopping and restarting ad nauseam?