Let $\mathbb{F}_p$ be a field of prime characteristic $p&gt;0$.The question is what we can say about the solvability of the equation $x^2+y^2=-1$ in $\mathbb{F}_p$ for every prime $p$. I have found out that there are $\frac{p-1}{2}$ different squares in the field $\mathbb{F}_p$. I was thinking about the circle of radius $kp-1$ has integer point for some $k$.
In an ordered field we have $x^2 \ge 0$, hence the equation $-1 = x^2 + y^2$ has no solution. But what about finite fields in general? What is the solutions set $$  -1 = x^2 + y^2 $$ of this equation?
How to show $\omega^n$ is a volume form in a symplectic manifold $(M, \omega)$? How to run 32-bit app in Ubuntu 64-bit? Getting incorrect slope values from an ASTER DEM in QGIS? I have a DEM layer from ASTER satellite and I want to get the slope information from it.  I use terrain analysis > slope, but the output is so wrong. Why is that?  Here is the DEM:     And here is the "Slope" that is totally wrong. It should stretch the values from 0 - 89.9097 but it does not. How can I capitalize the first letter of each word in a string in Perl? Is there a way that I can configure Visual Studio 2008 to understand CamelCase?  Specifically, I'd like to be able to get Ctrl + right or left cursor to take me to a subsection of a variable or type name. i.e., if my cursor was at the start of this line: LongNamedExampleClass longNamed = new LongNamedExampleClass();  and I hit Ctrl + right, I'd like to jump to &quot;Named&quot; in the class name, rather than the start of the variable name.  If I hit it again, I'd like to go to &quot;Example&quot;.  I hope that makes sense. I used this navigational feature in Eclipse quite a lot, and find it difficult to unlearn something so ingrained. Lebesgue measurable set that is not a Borel measurable set If a Matrix Has Only Zero as an Eigen-Value Then It Is Nilpotent After the downfall of Voldemort, what happened to the Death Eaters?  They committed many crimes, including mass murder and torture. Nothing in the books made mention of their punishments. What happened? I have created several designs for a particular project and they are related, but I don't want to have to keep recreating any similar elements, such as shared materials or textures.  How can I use the existing data in a new Blender file? Convert track bike to geared Is there any way to convert a track bike to a geared bike? I have a track bike with only the rails for the wheel but I've moved out of the city and there are more hills where I am now living so would like to convert to gears if possible and rather than buy a new bike, save some cash. How can I make my ad hoc iPhone application's icon show up in iTunes? I've got an iPhone app with icon file Icon.png.  This icon shows up properly when the app is on the phone itself, but it doesn't show up in the applications pane in iTunes.  What do I need to do to get it to show up properly? Geometric proof for irrationality of $\pi$ Is there a geometric proof for irrationality of $\pi$? That would be neat. Do I cast the result of malloc? In , someone suggested in a  that I should not cast the result of malloc. i.e., I should do this: int *sieve = malloc(sizeof(int) * length);  rather than: int *sieve = (int *) malloc(sizeof(int) * length);  Why would this be the case? Energia/MSP-EXP432 Issue: Analog In Pin removing voltage from Voltage Divider In the circuit below, I want to measure the voltage going to the ADC (output from the voltage divider consisting of a fixed resistor and photoresistor) using an MSP-EXP432; however, the MSP-EXP432 "Analog In" pin is lowering the output voltage (i.e. the voltage going into the ADC from the voltage divider drops when the "Analog In" pin is connected).  I set the analog pin as an input in the code shown below in Energia.  Is there any way to fix this?    const int bitsOfADC = 14; // 14-bit ADC Launchpad //int maxVal = 2^N -1;  // 2^14-1 int maxVal = 16383; float FSV = 5.0; // Supply to 0804ADC float LaunchPadResolution = FSV/maxVal; int D7 = A13; int D6 = A14; int ADCInput = A6; boolean D7_State = LOW; boolean D6_State = LOW; void setup() {   delay(1000);   Serial.begin(9600);   pinMode(D7, INPUT);   pinMode(D6, INPUT);   analogReadResolution(bitsOfADC);   pinMode(ADCInput, INPUT); } I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 How can I identify an anonymous inner class in a NotSerializableException Can I use the Google font 'Lato' for a logo design? I am creating a logo and would like to use the Google Font 'Lato', can I use this font in my logo design or are there copyright/limitations I need to worry about? I can't find anything online other than that it is an open source font that can be used in anyway you want (maybe that's my answer but I just want to make sure). Also, is it a bad idea to use a web-based font for a logo? "Protect Questions" privilege page has confusing wording The wording on the  privilege page is confusing and would benefit from being clarified. See these questions on  and . Suggested wording from  is to replace this:  Which questions can be protected? Any question, even a deleted question, can be protected and unprotected, subject to the following restrictions:  The question must be at least a day old A user can unprotect any protected question   with the following:  Which questions can be protected? Any question at least a day old, even a deleted question, can be protected and unprotected. A user with the privilege can unprotect any protected question. I get this warning:     Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release   What does it mean? Really stuck with this, \begin{cases}  2tx'(t)-1=x^2t \\  x(t=1)=1 \\  \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :(