How to compute $75^{75^{75}}$ modulo $32$?  I tried:  It's $\mathrm{gcd}(75,32)=1$, so with Euler's phi function I get   $\varphi(32)=32(1-\frac{1}{2})=16$.  Then it's $75^{16}\equiv1 \ (\mathrm{mod} \ 32)$.   So I computed $75^{75}$ modulo $16$:  It's $\varphi(16)=16(1-\frac{1}{2})=8$  So $75^{75}\equiv 1 \ (\mathrm{mod} \ 8)$ and $75^{75^{75}} \equiv 1  \ (\mathrm{mod} \ 8)$  Then it's $75^{75^{75}}\equiv 75^1 \equiv 75 \ (\mathrm{mod} \ 32)$.  I'm not sure if this is correct. Is there something wrong in this calculation or can it be done like this?
What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$?  The level of powers is $2020$.  First there is no parenthesis so it means 3 power of 3 which is also power 3 and so on 2020 times  Second I think that we can use Fermat's Little Theorem but I don't know how, and maybe there is a better way.
Case-sensitive Google search Is there any way to make a Google search case-sensitive? Excel: I need a formula that will count months between two dates. If it's 1-15 it should be a half month. Any ideas? So I am trying to find a formula for excel. Bascically I have two dates and I need to know the number of months between them. But to add complexity, I need a rule in there that if the earlier date is the 1st to 15th it will be a whole month but if it is 16th to end of the month it will be a half month.  For example: Date 1: April 4, 2012 Date 2: December 31, 2012 Answer: Number of months would be 9 months  If Date 1 was April 16, 2012, with the same Date 2 of December 31, 2012, then the number of months would equal 8.5.  Can anyone help me with this one? You help would be greatly appreciated. many thanks Cardinality of the set of all (real) discontinuous functions This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6.     (Show that) The cardinality of the set of all discontinuous functions is $2^{2^{\aleph_0}}$. [Hint: Using exercise 2.5, show that $|\mathbb{R}^\mathbb{R}-C|=2^{2^{\aleph_0}}$ whenever $|C|\leq 2^{\aleph_0}$.]   Exercise 2.5, as referred to in the hint, provides the following result:     For $n&gt;0$, $n \cdot 2^{2^{\aleph_0}} = \aleph_0 \cdot 2^{2^{\aleph_0}} = 2^{\aleph_0} \cdot 2^{2^{\aleph_0}} = 2^{2^{\aleph_0}} \cdot 2^{2^{\aleph_0}} = (2^{2^{\aleph_0}})^n = (2^{2^{\aleph_0}})^{\aleph_0} = (2^{2^{\aleph_0}})^{2^{\aleph_0}}=2^{2^{\aleph_0}}$.   The only way I could answer this question was by making use of two theorems stated later  in the same textbook. First, it is easy to prove that $|\mathbb{R}^\mathbb{R}-C|$ is an infinite set, since if it was finite we would have \begin{equation}2^{2^{\aleph_0}}=|\mathbb{R}^\mathbb{R}|=|\mathbb{R}^\mathbb{R}-C|+|C| \leq n+2^{\aleph_0}=2^{\aleph_0}&lt; 2^{2^{\aleph_0}},\end{equation}which is a contradiction. Now I make use of a theorem requiring the axiom of choice: For every infinite set $S$ there exists a unique aleph $\aleph_\alpha$ such that $|S|=\aleph_\alpha$. So I let $|\mathbb{R}^\mathbb{R}-C|=\aleph_\alpha$ for some ordinal $\alpha$. Now assume $\aleph_\alpha &lt; 2^{2^{\aleph_0}}$. Here I make use of another theorem, which i am not sure I am allowed to do: For every $\alpha$ and $\beta$ such that $\alpha \leq \beta$, we have $\aleph_\alpha+\aleph_\beta=\aleph_\beta$. So I basically then say either $|\mathbb{R}^\mathbb{R}-C|+|C|=\aleph_\alpha$ or $|\mathbb{R}^\mathbb{R}-C|+|C|=2^{\aleph_0}$, depending on whether $\aleph_\alpha \leq 2^{\aleph_0}$ or vice-versa. The assumption I am making is that $2^{\aleph_0}=\aleph_1$, which as I understand it is basically equivalent to the continuum hypothesis, as we are saying that the least uncountable number is $\aleph_1$ (but by the previous theorem requiring the axiom of choice this seems reasonable?). So anyway assuming I can do that, we again have a contradiction since then we are saying $|\mathbb{R}^\mathbb{R}|=2^{\aleph_0}&lt;2^{2^{\aleph_0}}$ or $|\mathbb{R}^\mathbb{R}|=\aleph_\alpha &lt; 2^{2^{\aleph_0}}$.  My main problem with my proof is that I am not making use of the result given in exercise 2.5 as suggested in the hint. Can anyone help with a proof making use of this result (stated in the second block quote above). Why is there a B in my flair? This was originally noticed and fixed back around the beginning of July. As of August 1, this problem seems to have returned. New bug report is here:   I just happened to notice that my SO flair has some sort of a B icon right before my username (instead of the expected site icon) Snapshot of the unknown icon:    My current (live) flair for monitoring:  It currently showing up like this for all users on all sites, but not on Combined or Meta flairs. For example Jon Skeet flair on programmers:    Live flair: MWE needs to be compiled with LuaTeX to reproduce the problem:  \documentclass[11pt]{report}% \usepackage{amsmath,mathtools} \usepackage{breqn}                          \begin{document} $J^{\ast} = \text{JFK} \to \text{NO} \to \text{SFO}$ \[ J^{\ast} = \text{JFK} \to \text{NO} \to \text{SFO} \] \end{document}   Gives (no compile error or any warning) the following strange output    I have to use breqn package. If I remove it, then I get the expected output    Is there a way to work around this, so I can use breqn and get correct output?  TL 2015. filelist: (compiled with lualatex)  This is LuaTeX, Version beta-0.80.0.1 (TeX Live 2015) (rev 5863)   *File List*   report.cls    2014/09/29 v1.4h Standard LaTeX document class   size11.clo    2014/09/29 v1.4h Standard LaTeX file (size option)  amsmath.sty    2013/01/14 v2.14 AMS math features  amstext.sty    2000/06/29 v2.01   amsgen.sty    1999/11/30 v2.0   amsbsy.sty    1999/11/29 v1.2d   amsopn.sty    1999/12/14 v2.01 operator names mathtools.sty    2015/11/12 v1.18 mathematical typesetting tools   keyval.sty    2014/10/28 v1.15 key=value parser (DPC)     calc.sty    2014/10/28 v4.3 Infix arithmetic (KKT,FJ)  mhsetup.sty    2010/01/21 v1.2a programming setup (MH)    breqn.sty    2015/08/11 v0.98d Breaking equations    expl3.sty    2015/12/20 v6326 L3 programming layer (loader)  expl3-code.tex    2015/12/20 v6326 L3 programming layer  l3pdfmode.def    2015/11/11 v6250 L3 Experimental driver: PDF mode flexisym.sty    2015/08/11 v0.98d Make math characters macros   cmbase.sym    2007/12/19 v0.92 mathstyle.sty    2015/08/11 v0.98d Tracking mathstyle implicitly graphicx.sty    2014/10/28 v1.0g Enhanced LaTeX Graphics (DPC,SPQR) graphics.sty    2014/10/28 v1.0p Standard LaTeX Graphics (DPC,SPQR)     trig.sty    1999/03/16 v1.09 sin cos tan (DPC) graphics.cfg    2010/04/23 v1.9 graphics configuration of TeX Live   pdftex.def    2011/05/27 v0.06d Graphics/color for pdfTeX infwarerr.sty    2010/04/08 v1.3 Providing info/warning/error messages (HO)  ltxcmds.sty    2011/11/09 v1.22 LaTeX kernel commands for general use (HO) supp-pdf.mkii pdftexcmds.sty    2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO) ifluatex.sty    2010/03/01 v1.3 Provides the ifluatex switch (HO)    ifpdf.sty    2011/01/30 v2.3 Provides the ifpdf switch (HO) luatex-loader.sty    2010/03/09 v0.4 Lua module loader (HO) epstopdf-base.sty    2010/02/09 v2.5 Base part for package epstopdf   grfext.sty    2010/08/19 v1.1 Manage graphics extensions (HO) kvdefinekeys.sty    2011/04/07 v1.3 Define keys (HO) kvoptions.sty    2011/06/30 v3.11 Key value format for package options (HO) kvsetkeys.sty    2012/04/25 v1.16 Key value parser (HO) etexcmds.sty    2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) epstopdf-sys.cfg    2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live  ***********   Update  I found another case of duplication when using breqn, which does not involve \text at all. But the solution provided below also fixes this issue. I am adding this second case here for reference just in case it might be useful. Here is MWE  \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{amsmath,mathtools} \usepackage{breqn} \begin{document} \[ \underset{\mathbf{S}} {\operatorname{arg\,min}}      \sum_{i=1}^{k} \sum_{\mathbf x \in S_i}       \left\| \mathbf x - \boldsymbol\mu_i \right\|^2  \] \end{document}   The output is    After adding the patch in the answer below the correct output is now given, which is I have a computer (Asus T200TA) with 2G RAM and 32-bit UEFI [Yep, what a wonderful idea to put 32-bit UEFI!]. I have two options:   Install a 64-bit system, boot manually and install grub-efi-ia32. Drawback: 2G is not much, systems with such low RAM are said to run better on 32-bit systems. Install a 32-bit version. (but from my state of knowledge it's not trivial to boot)   Which option would be better? Is there a term for when just by saying that something is broken and showing it to someone fixes the issue? This seems to be a common enough occurrence that it would merit its own phrase. I imagine it might be some sort of subset of Murphy's Law. But it's specific enough that Murphy's Law doesn't quite address it.  Do you know when you are banging your head against a wall for hours trying to fix something? And then when you finally give up and try to get someone else to come fix it (or even just look at it to see that it's broken), it's working perfectly as if nothing had ever happened? And at this point, you're just trying to convince the other person that it was broken in the first place.  It'd at least take some of the frustration out of the experience if you could both just laugh about &lt;insert the phrase for it here&gt;. But as it is now, I basically have to go through the whole paragraph above to say it, and at that point the joke is kind of lost. I got pretty confused at that question... Can anyone help me understand this question better and/or drop some hints please? Thank you! Difference between generalized linear models & generalized linear mixed models Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$ How do I temporarily disable an app when there is no "Disable" option Should I include a link to my Stack Overflow profile on my résumé? In Stein's Fourier Analysis i'm having trouble Justifying the following calculations for the Fourier Coefficients in $(1.)$  $(1.)$  Let $f(\theta) = \theta$ for $ -\pi \leq \theta \leq \pi$ in the case when $n \neq 0$:  $$f(n)=\frac{1}{2 \pi}\int_{-\pi}^{\pi}\theta e^{-in \theta} d \theta = \frac{1}{2 \pi}[-\frac{\theta}{in}e^{-in \theta}]_{-\pi}^{\pi} + \frac{1}{2 \pi in} \int_{-\pi}^{\pi}e^{-in\theta}d \theta = \frac{(-1)^{n+1}}{in} $$  Then the Fourier Series of $f$ is given by:  $$ f(n) = \sum_{n \neq 0}  \frac{(-1)^{n+1}}{in}$$ How do I send an HTTP request in C# without using System.Net classes? Product of binomial coefficient as a basis How can I clip an image via a bezier-path? Apologies for this possibly too vague question, but are there some general guidelines to follow when rendering artificial 3D meshes together with an imported 2D image as the background?  In the attached Blender file (or rendered image), the cube looks so out-of-the-scene. Some reflection from the green background onto the cube should make the cube more realistic in the scene? I don't know how to put it in professional terms -- any suggestions are welcomed! Booking a flight without passport info I have been looking to book a bundle (hotel and flight) with Expedia to Puerto Vallarta MX. When I get to the booking section it does not ask for my passport number. Do they ask for my passport after I book? Or later when the date gets near? My passport is currently being renewed and I want to book my hotel and flight before they sell out but I am afraid it'll ask me for my passport number. Can anyone inform me about this situation? I have a Python program that uses the "threading" module. Once every second, my program starts a new thread that fetches some data from the web, and stores this data to my hard drive. I would like to use sqlite3 to store these results, but I can't get it to work. The issue seems to be about the following line:  conn = sqlite3.connect("mydatabase.db")    If I put this line of code inside each thread, I get an OperationalError telling me that the database file is locked. I guess this means that another thread has mydatabase.db open through a sqlite3 connection and has locked it. If I put this line of code in the main program and pass the connection object (conn) to each thread, I get a ProgrammingError, saying that SQLite objects created in a thread can only be used in that same thread.   Previously I was storing all my results in CSV files, and did not have any of these file-locking issues. Hopefully this will be possible with sqlite. Any ideas? Does this polynomial evaluate to prime number whenever $x$ is a natural number?