For $p\geq5, 1^2+2^2+3^2+...+(p-1)^2$ is divisible by $p$.
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
Learn QM algebraic formulations and interpretations I have a good undergrad knowledge of quantum mechanics, and I'm interesting in reading up more about interpretation and in particular things related to how QM emerges algebraically from some reasonable real world assumptions. However I want to avoid the meticulous maths style and rather read something more meant for physicists (where rigorous proofs aren't needed and things are well-behaved ;) ) I.e. I'd prefer more intuitive resources as opposed to the rigorous texts.       Can you recommend some reading to get started? Long rectangular quads? Or neat squarish quads with tris? For curved meshes I've often noticed that in my desperation to avoid tris, I tend to extrude the mesh in a way which creates these long rectangular quads:    In this case at least, the mesh is all quads, however the edge flow causes loopcutting to be awkward in some places:    I tried creating the same mesh with more even topology, however I found tris unavoidable. Maybe it was because I was starting with the long-quad mesh, but I also found myself using the knife tool for everything..    This makes loopcuts less necessary and easier to place in most cases, though the occasional tri does get in the way.  An annoying thing about the tris are the poles, though they aren't too noticeable in this particular case.    Both techniques seems to have their advantages and their disadvantages:    Which is considered "better", if any? Are there any better workflows for creating even topology with as few tris as possible? Or is there some trick I'm missing altogether? Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? Does a car use friction to move? How do I add .e00 data into my QGIS project? Fighting Spam - What can I do as an: Email Administrator, Domain Owner, or User? Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? How do I escape the wildcard/asterisk character in bash? For example:  me$ FOO="BAR * BAR" me$ echo $FOO BAR file1 file2 file3 file4 BAR   and using the \ escape character:  me$ FOO="BAR \* BAR" me$ echo $FOO BAR \* BAR   I'm obviously doing something stupid.  How do I get the output BAR * BAR? What is the difference between re.search and re.match? What is the difference between the search() and match() functions in the ?  I've read the  (), but I never seem to remember it.  I keep having to look it up and re-learn it.  I'm hoping that someone will answer it clearly with examples so that (perhaps) it will stick in my head.  Or at least I'll have a better place to return with my question and it will take less time to re-learn it. I'm studying molecular biology and I'm trying to understand an experiment which shows the importance of promoters in the relative transcription level (RT). The image below comes from Rolf Knippers' book "Molekulare Genetik" (8th edition).    The legend says (among other things):     Die erste Zeile gibt die normale "Wildtyp"-Sequenz der 5'-flankierenden Region wieder.   Which, in english, means something like:     The first line repeats the normal wild-type sequence of the 5' flanking region.   The column on the right gives the relative transcription (RT) level, 1.0 being the highest possible level of transcription. As we can see, the lines where some parts of the "5' region" have been deleted give quite low RT levels, since some regions of the promoter are missing.  My questions are the following:  1) According to  and , I understand that RNA polymerase reads and uses both coding and non-coding strands in order to synthesize RNA. Therefore, how did they manage to use the sequences strands regions had been deleted to make a polymerase "read them" and perform transcription?  2) If the polymerase reads the template strand in the 3' -> 5' sense, shouldn't we talk of "ATAT box" or "TAAC box" instead of "TATA" or "CAAT" boxes? Does it mean that the "promoter" regions they are using on the graphic above are actually on the coding strand?  Thanks a lot for your help. Unable to mount /dev/loop0 during install What does Morgoth look like? Is Morgoth, the original enemy in J. R. R. Tolkien's Middle-earth ever visually described? I envisage him as a bigger, meaner version of the physical Sauron, but I cannot find any reference in The Simarillion. Idiom or expression that would imply "someone is a habitual liar" I'm looking for an idiom or expression for describing people who tell lies out of habit (so you cannot trust them at all).   I know that these people are called , but I'm not looking for psychological terms.  Is there any idiom or expression for describing such people?   For example:     "Did X tell that to you?! Come on, don't believe him/her. Don't you know he/she ___ (=is a habitual liar)?" How do I move my website to a new server without interruption to email? Fantasy Short Story that appeared in an anthology about 1968-1971 I read this short story around 1971.  The basic plot is for a steward to lead the king's son to another city state.  Along the way they have a number of interactions with people/subjects that demonstrate the immaturity of the son.  Last one involves the son beating up the suitor of a innkeeper's daughter.  When confronted by the steward, the son displays his contempt and chauvinistic regard of his future subjects and the steward kills him.  Turns out the trip was a test of his suitability to assume the crown and the steward explains to the king that he was not worthy.  Been looking in all the anthologies can seem unable to find it.  Thanks for your help Retrieving Maxwell's equations from the minimum action principle I'm trying to answer the following question from the book :  Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that   $E\bigg[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \bigg] \le CK$,  where $K = \max_i \|X_i\|_{\psi_2}$. Deduce that for ever $N \ge 2$ we have   $E\bigg[ \max_{i \le N} |X_i| \bigg] \le CK \sqrt{\log N}$.  I've tried to figure out what is the distribution of the maximum of Gaussians, but I'm reaching only inequalities that that don't help me answer the question.   I've also seen a similar question .  Does anyone have a clue or something to start with in order to answer this question?  Thanks! Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality  using the SB theorem? Also how does one find a bijection on $f:(0, \infty) \to (0,1)$ such that they have the same cardiniality? Questions link almost unreadable when active on meta If you click on  the font color is almost unreadable on the main site here.  Is it possible to make it a slightly different color? (I cannot say if darker or lighter is better)