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I've been working through a bit of Ireland and Rosen's Number Theory for fun. Problem 4.11 of Ireland and Rosen asks     Prove that $1^k+2^k+\cdots+(p-1)^k\equiv 0\pmod{p}$ if $p-1\nmid k$, and $-1\pmod{p}$ if $p-1\mid k$.    My work is leading where I didn't expect to go. Here I assume $p$ id odd. First, if $p-1\mid k$, then we can wrike $k=(p-1)j$ for some $j$. But then $$ 1^k+\cdots+(p-1)^k\equiv 1^{(p-1)j}+\cdots+(p-1)^{(p-1)j}\equiv \underbrace{1+\cdots+1}_{p-1\text{ times}}=\frac{(p-1)p}{2}\equiv 0\pmod{p} $$ since $p-1$ is even.   Secondly, I suppose $p-1\nmid k$. WLOG, I can assume $0&lt;k&lt;p-1$, Since if $k=(p-1)j+r$ with $0&lt;r&lt;p-1$, then  $$ a^k\equiv a^{(p-1)j+r}\equiv a^r\pmod{p}. $$  Then if $g$ is a primitive root, $$ 1^k+\cdots+(p-1)^k=\sum_{i=1}^{p-1}(g^k)^i=\frac{g^k(1-g^{k(p-1)})}{1-g^k}. $$ Since $g$ is primitive, $p\nmid 1-g^k$, but $p\mid 1-g^{k(p-1)}$, so the above sum is again congruent to $0\pmod{p}$.  So in all cases, I get the sum is congruent to $0$ regardless. Have I messed up, or is there a typo? Thanks. Firstly, I apologise that this question is specific to two polynomials. I understand that this post will not help a lot of people, and for that I am sorry. I will make it up to you all by anwsering lots more questions in the future! However, as my lecturers enjoy time off on the weekend, I am unable to ask this to anyone else who would know what the heck I'm talking about. Sorry if my notation is not widely used, I can elaborate more if required. Please excuse my poor Latex too.  Let:  $B = Y^2 - X^3 + X$, and   $F = (X^2 + Y^2)^3 - 4X^2Y^2$.  I wish to find the intersection number of $B$ and $F$ at $P = (0,0)$  i.e $I(P,F \cap B). $  However, I want to use the following proposition to do this:  If $P$ is simple on $B$ then $I(P,F \cap B) = ord_P^B(F) $.  I will omit proving that $P$ is simple on $B$ as I hope it is clear to see it is.  I know that $I(P,F \cap B) &gt; M_p(F) * M_p(B) = 1 * 4 $, where $M_P(G)$ is the multiplicity of $P$ on $G$.  Now, consider the Image of $X$ and $Y$:  $K[X,Y] \longrightarrow \frac{K[X,Y]}{Y^2 - X^3 + X} \longrightarrow O_P(Y^2 - X^3 + X)$  $X \longrightarrow x$  $Y \longrightarrow y$  Where lower case denotes the image.  Then $ord_P^{Y^2 - X^3 + X}((X^2 + Y^2)^3 - 4X^2Y^2)$  Can be found by substituting $y^2 = x^3 - x$ into $(x^2 + y^2)^3 - 4x^2y^2$  Which gives:  $ord_P^{Y^2 - X^3 + X}(x^9+3x^8 -5x^6 -4x^5 +3x^4 +3x^3)=3$  i.e $I(P,F \cap B) = 3$, a contradiction to  $I(P,F \cap B) &gt; 4$ as above.  The correct answer I know is 6, so where have I gone wrong? Determinant of a matrix whose $(i,j)$th element is $|i-j|$? Question states: "Let $D_n$ denote the determinant of the $n$-by-$n$ matrix whose $(i,j)$th element (the element in the $i$th row and $j$th column) is the absolute value of the difference between $i$ and $j$. Show that $D_n = (-1)^{n-1}(n-1)2^{n-2}$."  It seems like that I can get a recurrence relation (especially the question was from a book chapter dealing with recursion). In trying to get a recurrence relation, here is what I have done so far:  (Let $R_k$ denote the $k$th row and let $(i,j)$ denote the element/spot of $i$th row and $j$th spot)  1) replace $R_1$ with $R_1-(R_2+R_{n-1})$ to get $R_1^*$  2) replace $R_1^*$ with $R_1^*+R_{n-2}$  3) If I am not mistaken, I should have $-2$ on $(1,1)$, $2$ on both $(1,n-1)$ and $(1,n)$, and all other element in the first row as $0$.  4) take out $2$ from the first row, and expand the determinant about the first row  5) I should have $D_n=2(-D_{n-1}+(-1)^n B_{n-1})$, where $B_{n-1}$ is the determinant of a not-easily-reducible mayrix whose last column consists of $n-1$ ones followed by one $-1$.  How should I go about from now on? Once I get a recurrence relation, it would be pretty easy to get the explicit formula for $D_n$ using induction.  P.S. This question apparently is a question taken from 1969 Putnam Competition I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. Objects disappear when looking through camera Navigation bar overlapped by status bar Finite group with isomorphic normal subgroups and non-isomorphic quotients? I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? How to test randomness (case in point - Shuffling) First off, this question is ripped out from  question. I did it because I think this part is bigger than a sub-part of a longer question. If it offends, please pardon me.  Assume that you have a algorithm that generates randomness. Now how do you test it? Or to be more direct - Assume you have an algorithm that shuffles a deck of cards, how do you test that it's a perfectly random algorithm?  To add some theory to the problem - A deck of cards can be shuffled in 52! (52 factorial) different ways. Take a deck of cards, shuffle it by hand and write down the order of all cards. What is the probability that you would have gotten exactly that shuffle? Answer: 1 / 52!.  What is the chance that you, after shuffling, will get A, K, Q, J ... of each suit in a sequence? Answer 1 / 52!  So, just shuffling once and looking at the result will give you absolutely no information about your shuffling algorithms randomness. Twice and you have more information, Three even more...  How would you black box test a shuffling algorithm for randomness? How to close (kill) ssh ControlMaster connections manually Calculating sequential numbers into sorted table using ArcGIS Desktop? gdal.VectorTranslate returns an empty file So you are camping (please excuse the MMO jargon, sometimes I just can't help myself) your own particular corner of Stack Overflow, hoping for a couple of upvotes before bed.  It's freezing cold and you can't quite drag yourself across the room to grab the little blow heater, because that would mean leaving the almost adequate warmth of your laptop.  A completely objective question pops up, to which you absolutely know the answer.  Riding the surge of adrenaline, you spend the next twenty minutes typing like a madman. Desperate to beat the other lunatics that loiter in your tag of choice.  You beat them to the punch and enter a superlative answer, code samples, examples and maybe some ironic humour. You are flushed with your own awesomeness.  You go to bed. Satisfied you have garnered a couple of points of virtual self validation. You look forward to the flush of pleasure in the morning when you see your account bumped ever closer (yet never close enough) to that shining example of Skeetness.    Yet, the next morning, no such flush occurs.  Your answer sits in prime position for some questioner love and none eventuates.  You might get a couple of upvotes from some sympathetic SO compatriot, but the ultimate (I'll try to avoid any unsavory metaphors here)  satisfaction of accepted answer is denied you.    So my question. How you react to this kind of situation?   Catch yourself monitoring the questioner's account waiting for them to log on and then breaking nearby inanimate objects when your answer continues to be ignored. Panic that the questioner might delete their account, leaving your beloved question to evaporate in a puff of uninterest.  Wondering if it was something you said and answering some other of the questioner's queries in the hope of redeeming yourself in their fickle eyes.   I have seen a few examples of this type of frustration on Twitter and wondered how the hardcore Stack Overflow community dealt with this type of rejection. I'm Egyptian and flying to Cairo from Tbilisi, via Istanbul. The first flight Tbilisi-Istanbul is on Atlasjet, then from Istanbul to Cairo I am flying on Egyptair. For that, I have to recheck in at Istanbul to my Egyptair flight  Will I need to pass passport control at Istanbul airport to collect my bags and check in on Egyptair? Will I need a transit visa for that? Or it can be all done through neutral zone? Can I use apt-get or aptitude to install software to non standard location? For Christmas my kiddo got the D&amp;D Starter Set and we've been going though it and the full rules we downloaded online. I'm having a hard time figuring out how the example listed in the rules gets to a final value of 14.  The example is a level 1 character with wisdom of 15 and a proficiency in perception.  I understand that you start with  10 (base) + 2 (character level) + 2 (wisdom 15) which gets you to 14 but then how does the "proficiency in perception" help or factor in?   What gets more confusing is on one of the pre-generated character sheets has a level 1 character with wisdom of 10 but a passive perception value of 10. If I follow the math from above shouldn't it be 10 (base) + 2 (char lev 1) + 0 (wisdom 10 = 0 bonus) for a final score of 12?  How to calculate passive perception? Convert char to int in C# I have a char in c#:  char foo = '2';   Now I want to get the 2 into an int.  I find that Convert.ToInt32 returns the actual decimal value of the char and not the number 2.  The following will work:  int bar = Convert.ToInt32(new string(foo, 1));   int.parse only works on strings as well.    Is there no native function in C# to go from a char to int without making it a string?  I know this is trivial but it just seems odd that there's nothing native to directly make the conversion. What is the appropriate statistical test when calculating the busiest day of the week? We have 10 years of data and Monday is the busiest (most visits to the hospital) on average. If we wanted to assess whether that is true and report a p-value, what is the appropriate statistic?   We believe the answer is chi-square as we found a peer-reviewed medical article from the Annals of Emergency Medicine (highest circulating journal for that specialty) that used chi-square for this very calculation.   If chi-square is appropriate, what values do we use for observed and expected values for chi-square? One approach would be to list all the values for Monday as observed values, or all the means for Mondays averaged across a month, or all the means for Mondays averaged across a year.  If there were no busiest or least busy days, we would assume that the visits would be the same every day of the week. That means our "expected values" for Chi-Square would be the mean of the visits for any given week. In a reverse split, what happens to odd lots? Let us say for example that I have 27 shares of stock X (there really is an X but its identity is not relevant).  In this example, X does a 5:1 reverse split. How many shares do I end up with. If I have fractional shares, what can I do with them? Can you take the same feat multiple times? Electricity takes the path of least resistance? Electricity takes the path of least resistance!   Is this statement correct?    If so, why is it the case?  If there are two paths available, and one, for example, has a resistor, why would the current run through the other path only, and not both?