I noticed that for elliptic curves of the form $y^2≡x^3+a\pmod p$ sometimes the number of points is always $p+1$ for any choice of $a$. This seems to be the case for all $p ≡ 5 \pmod 6$. Moreover, when this does not happen, i.e., for $p ≡ 1 \pmod 6$, it looks like there are exactly zero curves of such form where the number of points is $p+1$. Can someone point me towards the right direction as to why this happens?
Let $p &gt; 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points.  Using Fermat's little theorem you can prove that $x^3 + 1$ is a bijection on $\mathbb{F}_p$. Hence, $$\#E(\mathbb{F}_p) = \#\{(x,y) \in \mathbb{F}_p^2 : y^2 = x^3 + 1\} + \#\{\infty\} = \#\{(x,y) \in \mathbb{F}_p^2 : y^2 = x\} + 1.$$ But then I am stuck trying to prove that $y^2 = x$ has $p$ solutions $(x,y) \in \mathbb{F}_p^2$.
Book: Nanobot apocalypse on Earth, originally used for weather control A few years ago I read an excellent sf book about nanobots that went out of control. In the book they were used for weather control, when they went amok the scientists threw in yet another bunch of nanites and so on, until Earth became uninhabitable, teeming with nanites, I remember the weather also became cold. The planet still could be visited but when they lost cloaking the nanites would attack.   Also, some humans in the book merged with nanites, they could behave benevolently but I forgot the plot too. I only remember fragments of this book, I would like to read it again in English. (loaned a long time ago from a library here in Sweden but they closed the library).   I only know that it was written by an Anglo-American contemporary SF writer, probably 30-55 y.o. What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? /A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? Is Innate Spellcasting counterspellable? On p.85 of Xanathar’s Guide to Everything, there’s a section clarifying that you need to perceive a spell being cast (by either a verbal, somatic or material component) in order to counterspell it.  It then goes on saying this :     If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible.   The thing is that many monsters’ Innate spellcasting specify that only material components are discarded (not verbal or somatic). So I am not sure whether you can actually counterspell innate spellcasting when the stat block only mentions removing material components.  There seems to be  supporting that such spells could be counterspelled (although it is confusing a bit).  It basically comes down to whether the above quote’s “many creatures” refers to all creatures with innate spellcasting, or only those that forgo all spell components to innanely cast. So, which is it ? I just bought "Reputation to the max". I bought this first, because I really want to help the cleanup of the vote-to-close queue, which, despite the efforts I've been hearing about, is still quite large. According to SO, my reputation is now 15,750, but that queue still says I need at least 3k rep to review! That is clearly a bug! How do ideal quotients behave with respect to localization? Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When does localization at $S$ commute with taking quotients, i.e. when does the equality $$(S^{-1} J : S^{-1} I) = S^{-1} (J:I)$$ hold? More generally, when is it true that $$S^{-1} \text{Ann}_{R} (M) = \text{Ann}_{S^{-1} R} (S^{-1} M)$$ for an $R$-module $M$? Dell Inspiron 3551 Unresponsive Touchpad I just got my laptop in the mail and I installed Ubuntu 14.04 LTS. It all went smooth until it booted up and I noticed my touch pad doesn’t work at all. It works perfectly on Windows and when I use a wireless mouse it works as well.  Here's my xinput:  ⎡ Virtual core pointer                          id=2    [master pointer  (3)]   ⎜   ↳ Virtual core XTEST pointer                    id=4    [slave  pointer  (2)]   ⎜   ↳ 2.4GHz RF  KEYBOARD  AND  MOUSE               id=11   [slave  pointer  (2)]   ⎜   ↳ DLL06AB:00 06CB:78F1                          id=13   [slave  pointer  (2)]   ⎜   ↳ SynPS/2 Synaptics TouchPad                    id=15   [slave  pointer  (2)]   ⎣ Virtual core keyboard                         id=3    [master keyboard (2)]       ↳ Virtual core XTEST keyboard                   id=5    [slave  keyboard (3)]       ↳ Power Button                                  id=6    [slave  keyboard (3)]       ↳ Video Bus                                     id=7    [slave  keyboard (3)]       ↳ Power Button                                  id=8    [slave  keyboard (3)]       ↳ Sleep Button                                  id=9    [slave  keyboard (3)]       ↳ 2.4GHz RF  KEYBOARD  AND  MOUSE               id=10   [slave  keyboard (3)]       ↳ Integrated_Webcam_HD                          id=12   [slave  keyboard (3)]       ↳ AT Translated Set 2 keyboard                  id=14   [slave  keyboard (3)]       ↳ Dell WMI hotkeys                              id=16   [slave  keyboard (3)]     I would love to have my track pad working and I've done some googling and browsing of this site and nothing I've tried has worked. Any input would be great. Let me know if there’s any other information that can help remedy this. I'm pretty new to Linux. Thank you very much.  Edit: I've updated the kernel to 3.19 and checked my Additional Drivers which had nothing for me. Neither of which worked. Can a spell be prepared once and cast multiple times? Unresolved external symbol on static class members How to lead a moving target from a moving shooter I saw this question: .  My situation is a little different though.   My target moves, and the shooter moves.  Also, the shooter's velocity is added to the bullets' velocities, i.e. bullets fired while sliding to the right will have a greater velocity toward the right.   What I'm trying to do is to get the enemy to be able to determine where they need to shoot in order to hit the player.  Using the linked SO solution, unless the player and enemy are stationary, the velocity difference will cause a miss.  How can I prevent that?      Here is the solution presented from the stack overflow answer.  It boils down to solving a quadratic equation of the form:  a * sqr(x) + b * x + c == 0   Note that by sqr I mean square, as opposed to square root. Use the following values:  a := sqr(target.velocityX) + sqr(target.velocityY) - sqr(projectile_speed) b := 2 * (target.velocityX * (target.startX - cannon.X)       + target.velocityY * (target.startY - cannon.Y)) c := sqr(target.startX - cannon.X) + sqr(target.startY - cannon.Y)   Now we can look at the discriminant to determine if we have a possible solution.  disc := sqr(b) - 4 * a * c   If the discriminant is less than 0, forget about hitting your target -- your projectile can never get there in time. Otherwise, look at two candidate solutions:  t1 := (-b + sqrt(disc)) / (2 * a) t2 := (-b - sqrt(disc)) / (2 * a)   Note that if disc == 0 then t1 and t2 are equal. Using the expression "the same" for a previously mentioned item We are doing the research about applying remote sensed data on hydro logical model . After delineating, the river basin from DEM data using Argis9.3, We have to map the remote sensed data derived from products like PERSIANN. Our task now is to create the grid with the same resolution with the available remote sensed data( for example 0.25degree x 0.25 degree). and have a certain corner that matches with the obtained data.   Does anyone know how to deal with our problem? Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . Is it common to use "where" after "a time" as a relative? I heard the following comment of Mr. Marco Rubio about Mr. Donald Trump on CBS’s “Face the Nation” in a clip in February 29th AP Radio News:  “There is never gonna be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald Trump. Donald Trump is not Republican Conservative. Donald Trump is trying to pull off the biggest scam in American political history.”  I thought I misheard “There is never gonna be a time where –“ for “There is never gonna be a time when –“ at first, because I was in understanding that time should be trailed by “when” as a relative noun or adverb. An English Japanese dictionary at hand (Readers English Japanese Dictionary published by Kenkyusha) provides “where” singly as a relative to refer the ‘place.’ However, my lookup of the following two sources endorsed that I heard it right:  Here is what is never going to happen in this race. There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to get out for purposes of rallying around Donald..  “Here is what is never going to happen,” Rubio continued. “There's never going to be a time where the Republican Party rallies around and says you have to get out or anyone has to ...   Is it common, at least not uncommon to use “where” after “time” as a relative? Determine the exact value of equations involving more two trig variables $2\cos^2x=1+\sin x$.  Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$.    I am experiencing problems with factoring this question.  First I started by getting everything on to the left side so: $2\cos^2x-\sin x-1=0$.  Then seeing as there are two trig variable here ($\cos$ and $\sin$), I used the pythagorean identity and replaced $2\cos^2x$ with $1-2\sin^2x$ to make it into a one variable question: $1-1-2\sin^2x-\sin x-1=0$.   By collecting like terms it then becomes $-\sin^2x-\sin x=0$.  Now you could factor these with $0$ and $-1$, but then it creates uneven brackets so this musn't work.  I don't think the quadratic formula is necessary, but I am wondering if this could be treated as a difference of squares question.    If you replace the $\sin x$ and make the question into $-x^2-x=0$ it is a little easier to look at, however I am not sure if this is the right approach.  If someone could please clarify my answer, that would be appreciated! I guess that energy will be used up and, at the end, will contribute to heat the earth, so I see no big differences...  please explain your point of view. How to break out of multiple loops? Given the following code (that doesn't work):  while True:     #snip: print out current state     while True:         ok = get_input("Is this ok? (y/n)")         if ok.lower() == "y": break 2 #this doesn't work :(         if ok.lower() == "n": break     #do more processing with menus and stuff   Is there a way to make this work? Or do I have do one check to break out of the input loop, then another, more limited, check in the outside loop to break out all together if the user is satisfied? Increase iPhone call history How can see call history in iPhone one month back, i also tried with tried party software but that is not perfect , is the any solution from apple? I have a WinForms app that I am trying to make full screen (somewhat like what VS does in full screen mode).  Currently I am setting FormBorderStyle to None and WindowState to Maximized which gives me a little more space, but it doesn't cover over the taskbar if it is visible.  What do I need to do to use that space as well?  For bonus points, is there something I can do to make my MenuStrip autohide to give up that space as well? In this video, a college basketball player steals the ball, races down court, and avoids the defender by throwing a pass to himself off the backboard.    In all the (disorganized) basketball games I've played in, a shot had to hit the rim before the shooter was allowed to rebound. And that looks like a shot to me.  Obviously the officials allowed it, but I couldn't find an explicit mention in the . Is this ruled as a shot, and if so, why is it legal? I'd be interested in the answer for any level of basketball, though NCAA is the most relevant. When I went to the review overview on Meta on 08.01.2015, 19:16, it listed two open items in the suggested-edits queue, however when I clicked on the queue, I was told that the queue had been cleared. When I went back to the review overview, however, the edits were there again.  Since the icons for recent reviews had not moved, I suppose that these were still the same two items waiting for review, and since phenomenon lasted for half an hour now, I would rule out  or  to be the reason for this.  After a while, the phenomenon ceased, with one new reviewer popping up in the icon list of recent reviewers.  I had not performed or skipped any suggested-edit review from the queue at that point.    A similar phenomenon occurred for several hours up to 11.04.2015, 11:05: The review overview showed two suggested edits and when I went to the suggested-edits queue, it was cleared. Sometimes, edits that I could review appeared, but two phantom edits remained.