Original problem:  If $\alpha$ be the greatest positive root of $x^3-3x^2+1=0$, then show that $[\alpha^{1996}]$ is divisible by 17. ($[x]$ denotes the greatest integer less than or equal to $x$)  My work: It is easily found that if the roots are $\gamma &lt; \beta &lt;\alpha$, then $-1 &lt; \gamma &lt; 0 &lt; \beta &lt; 1$ and $2 &lt; \alpha &lt; 3$, and that they must be irrational. We also have, $\begin{align*} \alpha + \beta + \gamma &amp;= 3 \\ \alpha\beta + \beta\gamma + \gamma\alpha &amp;= 0 \\ \alpha\beta\gamma &amp;= -1 \\ \alpha^2 + \beta^2 + \gamma^2 &amp;= 9 \\ \alpha^4 + \beta^4 + \gamma^4 &amp;= 81-12 = 69 \equiv 1 \pmod{17}\\ \end{align*}$ Now, we also note that $-\beta &lt; \gamma &lt; 0 &lt; \beta &lt; 0.9$, and using the inequality $(1+x)^r \leq e^{rx}$, we have that $0 &lt; \beta^{1996} + \gamma^{1996} &lt; 0.5$. From Newton's identities, we know that $\alpha^{1996} + \beta^{1996} + \gamma^{1996} \in \mathbb{Z}$. Hence, from the definition of the greatest integer function, $$[\alpha^{1996}] = \alpha^{1996} + \beta^{1996} + \gamma^{1996} = \left(\alpha^4\right)^{499} + \left(\beta^4\right)^{499} + \left(\gamma^4\right)^{499}$$ But we have that $\alpha^4 + \beta^4 + \gamma^4 \equiv 1 \pmod{17} \implies (\alpha^4 + \beta^4 + \gamma^4)^{499} \equiv 1 \pmod{17}$. To prove the required result, we must show that the rest of the terms of the expansion must sum to a multiple of 17. Although we know that the sum must be an integer, as the terms themselves are irrational, I can't think of any way to show this, or even if it is true or not. Any help or hint regarding this will be appreciated. I am also grateful to any hints on how to solve the original problem in any other easier way.
Let $\alpha,\beta,\gamma$ be the roots of $x^3-3x^2+1 = 0$. Prove that $$\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$$ where $n$ is an integer.   It sort of makes sense why they are congruent since we can say $x^3-3x^2+1 \equiv (x-4)(x-5)(x+6) \pmod{17}$, but since $\alpha,\beta,\gamma$ aren't integers, how do we prove this?
What is a word for a person with more expertise (in certain field) than an intermediate but less so than an expert? Annoying space before the first item in an enumeration which starts a theorem Consider the following example:  \documentclass{article} \newtheorem{Theorem}{Theorem} \begin{document} \begin{Theorem} On the other hand, this one has only one claim. \end{Theorem} \begin{Theorem} \begin{enumerate} \item First claim \item Second claim \end{enumerate} \end{Theorem} \end{document}   Typesetting it results in:    Notice there is extra horizontal space before the first item in the list of the second theorem.     Is there a sensible way to get rid of that?   Ideally, the solution would work with amsthm, thmtools, enumitem.  N.B.: this is related to  but not quite the same. Why does input capacity of an opamp increase noise gain? On the image below you can see an inverting amplifier with input capacity C1 and capacity C_F for general stability.     Now my book shows me following frequency plot:  A_n ... Noise-Gain  A_D ... open-loop gain  A_U ... V_out / V_in --> inverting amplifier gain     You can clearly see that C1 is increasing the noise gain at some point. And C_f is stabilizing A_n and A_U.   But why exactly does C1 increase the noise gain exactly? I need to convert some files to PDF and then attach them to an email.  I'm using Pear Mail for the email side of it and that's fine (mostly--still working out some issues) but as part of this I need to create temporary files.  Now I could use the  function but it sounds like it creates a file on the filesystem, which isn't what I want.  I just want a name in the temporary file system (using ) that won't clash with someone else running the same script of the same user invoking the script more than once.  Suggestions? error: file '/grub/i386-pc/normal.mod' not found. grub rescue&gt;   What can I do? I just sit and stare at it.   I found my old netbook (Dell Inspiron 1010) which I have not used for about four years. I replaced Windows XP with Ubuntu 12.10. I used my bootable USB drive. I installed and rebooted. I got the message that normal.mod is not found.   What should I do? Type exit, reboot, or quit? Should I re-install? I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.  Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by :     New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells. I have a webpage that implements a set of tabs each showing different content. The tab clicks do not refresh the page but hide/unhide contents at the client side.  Now there is a requirement to change the page title according to the tab selected on the page ( for SEO reasons ). Is this possible? Can someone suggest a solution to dynamically alter the page title via javascript without reloading the page? Solder new RAM in a 2015 Retina MacBook Pro I know that the new rMBPs have no "user-serviceable" RAM. But is it possible to solder a new RAM chip onto a 2015 rMBP? Given that the technician doing this is able to install the new chip without damaging anything, would it work? What ultimates can be damage boosted by Mercy? What ultimates can be damage boosted by Mercy?  We already know that , but are there other ultimates that can be damage boosted by Mercy? Can all ultimates be damage boosted by Mercy? Why must a singularity form inside a black hole? How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?  It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it should be easy. There is a hint that says to first prove when $m$ and $n$ are coprimes. I did this part. But I have no idea how to solve the general case.  Thanks. LCM. What am I missing? Problem statement:  Let $S$ be the set of all positive integers $n$ such that $n^2$ is a multiple of both $24$ and $108$. Which of the following integers are divisors of every integer $n$ in $S$? The choices are: $12, 24, 36, 72$. The solution is $12$ and $36$. My solution was $S = \{36, 36* 6^3, 36*6^5, ...\}$  I can't figure out this problem. I started by finding the $\text{lcm}(24,108)$ and solved for $n$. I got some of the numbers in the set $S$, but not all of them, and I don't see what I'm missing. Update: added the choices and what my results One frequent aspect of pdflatex output that could be improved is to what extent one can select text output from the generated pdf-file and copy-and-paste it. This is essentially an issue of having the right output encoding. There are at least two important aspects to this:   In the ideal case, Unicode would be the output encoding. Ligature glyphs (such as ﬁ) are ideally de-ligated in the output (here: "fi").   It seems that output encodings are specified through the fontenc package. What is the range of possible output encodings, and most importantly: Is there a way to specify a Unicode output encoding and one that deals with ligatures in the intended way?  (Note: It is important to distinguish output from input encodings. How to use different input encodings in LaTeX has been documented widely. My understanding is that this is best handled by the inputenx package, as it supersedes the inputenc package.)    Quick guide to the solutions: The information in the answers and answer threads is a bit distributed, so here the short summary: one approach is to use \input glyphtounicode together with \pdfgentounicode=1; the other approach is to use cmap/mmap.  Addendum: Sometimes one will need to load the package accsupp and enclose one's macro definition in \BeginAccSupp{method=hex,unicode,ActualText=&lt;codepoint&gt;}[...]\EndAccSupp{} to generate a specific code point. See the caveat about non-BMP code points  and the fix (starting from accsupp version v0.4, 2012/11/18) , which provides the new unichar package option. Organize uploads by year, month and day How OPTIONS are settled can be an eye opener and I hope I can get some answers The only reason why I put it here is to find out whether I am a victim or just ignorant of the way brokers deal with option positions that were exercised. Whether I am a victim or just ignorant depends on the answer that I posted here. The facts are: I had a 2 SHORT QS Option at 75 which was exercised against me on the 23 Dec 00.45 AM and I later exercised my 2 LONG QS Option at 70 on 23 Dec 9.31 AM. The broker charged me Borrow Fee for 24-27 Dec for $1102, which means instead of getting $1000 profit I turn this into a loss of $102 plus comm. My question:  Since both long and short options were exercised on the same day one at 45 mins on 23 Dec and the later at 9.31 am on the same day, how can there be a SHORT POSITION in the stocks since both are net-off in my account ?  The answer I got from my broker and this is the point that I want to clarify.  The response I got from IB was that the initial exercise at 45 mins on 23 Dec was actually on 22 Dec (not 23 as I alleged). They refused to show me how they got 22 Dec.  This is my explanation to my Broker in reply.  Even if the short call and long call was exercised on different dates 22 and 23 Dec respectively, IB can only charge me for the SHORT position for one day and not from 24-27 Dec by ignoring my long call position was not there.  Further Facts:  My Broker ignored my answer and insisted the dates are correct and therefore I was charged to borrow shares that was need to satisfy the short position on exercised. So I am lost as I do not know whether the broker is correct and they can do this by ignoring my long call position after exercised or there is something they had not tell me. That is the reason I came to this site soliciting answers from those who know more than I do or had experienced the same.  I hope I have explained this clearly now and why would I want to give an opinion when I am looking for an answer ? The annual student intake of Hogwarts seems about 100. Since in book 7 Hogwarts is declared mandatory for all children, this means that the total magical population of Britain cannot be much more than 10,000.  How do these people make a living?  All jobs mentioned in the books are non-productive occupations (government, stores, sports &amp;c) - not farmers or engineers. There are exceptions - makers of wands/robes/broomsticks - but where do they get the raw materials from?  Who and how "creates goods" in the wizarding world?  Note that if you say "wizards exchange their gold for Muggle money and buy stuff from Muggles (through intermediaries, if necessary)" then you would have to explain where they get their gold: they would either have to mine it, or produce something of value to the Muggles (in which case the separate financial system of galleons/sickles/knuts does not make sense).  PS. A simple issue of maintaining a steam locomotive for Hogwarts express is pretty much insurmountable for 10k people - without Magic, that is. So, I can imagine that the skilled tradesmen can transfigure dirt into raw materials for their trade - except for food, which "is the first of the five Principal Exceptions to Gamp's Law of Elemental Transfiguration". HP PSC-2100 can't seem to get xsane and HPLIP (printing working) Probability of sampling with and without replacement In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.  Please help me show how this is proved. People are born and grow up and die of old age in a matter of 5 days. The elders (5 days old) train children and send the most promising one on a trek across the dessert to a monolith that is the cause of the rapid aging to hopefully disable it.   I saw it on a reel to reel in the mid 90's. It looks like it was made in the late 70's or 80's. The cast was small. A few men played the elders seated in a row of chairs and the young boy who had to set out to save them who matures into a man and finally an elderly man.  I've been trying to figure out the title for years. Difference between == and === in JavaScript