Is there any method that I can follow to calculate $x$ and $y$ such that $p= x^{2}+y^{2} $ where $p$ is a prime and $p \equiv  1 ({mod}\ 4)$. Fermats theorem of sum of two squares.
The web is littered with any number of pages () giving an existence and uniqueness proof that a pair of squares can be found summing to primes congruent to 1 mod 4 (and also that there are no such pairs for primes congruent to 3 mod 4).  However, none of the stuff I've read on the topic offers any help with actually efficiently finding (ie other than a straight search up to sqrt(p)) the concrete values of such squares.  What's the best way to actually find them ?
Why do both cron and systemd execute automatic upgrades? What does ` (backquote/backtick) mean in commands? Is there any evidence or reason why FIFA is so against adding instant replay to soccer? I find myself seeing many unlucky or just plain wrong decisions made by refs that could be corrected with instant replay. Will factory resetting my device remove the Android upgrade? I have upgraded my phone to Android 2.1. I need to reset my phone but I don't want to lose the upgrade. Is it possible to do this? How? If the laws of random variables weakly converge to a point mass at $c$, then the random variables converge in probability to $c$. This is question 10.3.1 from Rosenthal's First Look at Rigorous Probability.  Suppose $\mathcal{L}(X_n) \Rightarrow \delta_c$ for some $c \in \mathbb{R}$. Prove that $\{X_n\}$ converges to $c$ in probability.  Here $\mathcal{L}(X_n)$ refers to the distribution of $X_n$.  Is the following proof correct?  Is a simpler proof available?  Is there a simple proof using Skorohod's theorem?  I attempted to prove the contrapositive. Suppose $X_n \not \to c$ in probability.  This means that there is an $\varepsilon &gt; 0$ so that for all $N$, there exists an $m &gt; N$ such that  \begin{equation}   P(|X_m - c| \geq \varepsilon) &gt; \eta, \text{ for some $\eta &gt; 0$}. \end{equation}  Consider  \begin{equation}   B := \{x \in \mathbb{R} : |x - c| \leq \varepsilon\} \in \mathcal{B} \text{ the Borel sets on $\mathbb{R}$}. \end{equation}  Then $B^c \in \mathcal{B}$ and denoting the distribution of $X_n$ as $\mu_n$ we have  \begin{equation}   \mu_n(B^c) = P(X_n \in B^c) = P(|X_n - c| \geq \varepsilon) &gt; \eta. \end{equation}  For all $N$, there exists an $m &gt; N$ such that the cumulative distribution function of $X_m$ is not the same as the cumulative distribution function of $\delta_c$; in particular it's mass changes somewhere at least $\varepsilon$ distance from $c$. Finally because $\delta_c(\{x\}) = 0$ for all $x \neq c$ this completes the proof. In the following example, is it appropriate to use a or an as the indefinite article, and why?     He ate __ green apple.   I know that in the case of just "apple", it would be "an apple," but I've heard conflicting answers for "green apple," where the noun is separated from the article by an adjective.  Also, which is more appropriate in this case:     He ate  __ enormous Pop-Tart. A limit question (JEE $2014$) The following is a JEE (A national level entrance test) question:  Find the largest value of the non-negative integer ( a ) for which:  $$ \displaystyle \lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4 $$  On solving this, we get $ a = 0 $ or $ a = 2 $. So we take the answer as $ 2 $. But the official answer key says that the answer is $ 0 $. The reason given (not in the official key - they simple contain the answer. I saw the reason in an "unofficial" solution to the question paper) is that if $ a = 2 $, then the term $\displaystyle \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } $ tends to a negative value ( $-0.5$ ). So $ a = 0$.  BUT:  By putting $a = 2$ in the limit, and inputting that to wolframalpha (), the answer turns out to be $0.25$. So, by putting $ a = 2$ also, we get the same value of the limit - $0.25$.  How to solve this dispute? Also, could you please check with Mathematica? Unfortunately, I don't have access to it!  (This is very important as the change in the answer key will affect the rankings in a dramatic way) CSV for Excel, Including Both Leading Zeros and Commas Word with the same meaning as "good moments" or "pleasant memories"? Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. Why $¬¬\bot \not\in PROP$? I'm reading Dalen's Logic and Structure.  He gives the definition of the set $PROP$:     Definition $\bf2.1.2$ The set $PROP$ of propositions is the smallest set $X$ with the properties       $\begin{array}{rl} \rm(i)&amp;p_i\in X(i\in N),\bot\in X,\\ \rm(ii)&amp;\varphi,\psi\in X\Rightarrow(\varphi\wedge\psi),(\varphi\vee\psi),(\varphi\to\psi),(\varphi\leftrightarrow\psi)\in X,\\ \rm(iii)&amp;\varphi\in X\Rightarrow(\neg\varphi)\in X.\\ \end{array}$   Then there are some examples:        And he says that $¬¬\bot \not\in PROP$, but I don't understand it. Isn't $¬¬  \bot=\bot$ due to the property of double negation? Considering this is true, $\bot\in PROP$. Of course, in this question, I'm pressuposing some kind of evaluation, that $¬¬ \bot$ becomes $\bot$ due to the law of the double negation, but perhaps it should be hold in an unevaluated form.  Ex:. $1+1=2$ because we presupose the $1$ merges with $1$ and then becomes two. If the merge (evaluation) does not happen, then I guess:  $$1+1=1+1$$ $$2=2$$ $$2\neq 1+1$$ $$1+1\neq 2$$  Perhaps it's something like that, I took the idea from Mathematica. Sometimes it does treats things as different in the absence of full evaluation. Why does chmod 777 -R / leave the system unusable? I'm only granting permission to all to do anything but why does the system crash by giving permissions only? I'm only modifying the permission not changing the files. What does it mean if a Python object is "subscriptable" or not? In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?    Thanks for the answers. Several answers basically said for irrational numbers that  A Dedekind cut is a pair of sets of rational numbers $\{L, R\}$. The set of real numbers is defined to be the set of all Dedekind cuts, where a Dedekind cut is a pair of sets of rational numbers $\{L, R\}$ which have no elements in common, and where all the elements of $L$ are less than any element of $R$.  Each Dedekind cut is a real number. This is where I have a problem - surely that can’t be correct. The set $L$ is a set of all rationals, and there must be a rational in the set $L$ that is greater than all other rationals in that set, even if we have no method of determining it. And similarly, there must be a rational in the set $R$ that is less than all other rationals in that set, even if we have no method of determining it. If every irrational number has a corresponding set $L$, then each irrational number has some such corresponding largest element of that set $L$, and then each irrational number has some corresponding rational number. And that would mean that the irrational numbers are countable. So, with Dedekind cuts,  the only conclusion is that there must be irrational numbers $x$ which are either greater or lesser than some irrational cut $y$ of the rationals, and between $x$ and $y$ there is no rational number.  But that is impossible, so that the Dedekind cuts cannot be the correct representation of the real numbers.  Surely the problem with Dedekind cuts is in using sets of rationals that include all rationals up to a certain rational. But there is an alternative method of representing irrationals can be defined in terms of infinite sets of rational numbers. For example, in binary notation, the non-integer part of $\pi$ is $.00100100\ 00111111\ 01101010\ 10001$. You define a set by: if the nth digit is a $1$, then  the natural number $n$ is in the set. And then we have that, for the real numbers between $0$ and $1$, that the set of real numbers is simply the set of all subsets of natural numbers. Each subset corresponds to some real number between $0$ and $1$.  And in this way, all real numbers can be considered to be some set based only on nested sets of the empty set.  But I still haven’t got a satisfactory answer for how negative numbers can be represented in terms only of sets containing the empty set. Any ideas? Use of 'const' for function parameters How far do you go with const?  Do you just make functions const when necessary or do you go the whole hog and use it everywhere?  For example, imagine a simple mutator that takes a single boolean parameter:  void SetValue(const bool b) { my_val_ = b; }   Is that const actually useful?  Personally I opt to use it extensively, including parameters, but in this case I wonder if it's worthwhile?  I was also surprised to learn that you can omit const from parameters in a function declaration but can include it in the function definition, e.g.:  .h file  void func(int n, long l);   .cpp file  void func(const int n, const long l)   Is there a reason for this? It seems a little unusual to me. When I press the power button on my Epson WorkForce WF-3520 printer, it starts up normally, but shuts down within 30 seconds. Ten seconds later, it starts itself and the stop/start cycle is repeated again and again. This has only started happening today (6th December 2016). how to get the original entity on hook_entity_presave? i actually need it for the content_moderation_state entity   function hook_entity_presave(Drupal\Core\Entity\EntityInterface $entity) { if ($entity-&gt;getEntityTypeId() !== 'content_moderation_state' ) {     return; }     $moderation_state = $entity-&gt;get('moderation_state')-&gt;getString();   //code to get the original entity moderation state Can we get wordpress.stackexchange.com added to the list of off-topic redirection sites? I'm noticing a lot of WordPress questions being asked on Stack Overflow lately. I can't flag these to be moved to wordpress.stackexchange.com because it's not in the list:    Can it be added? Are there any other Stack Exchange sites worth adding? I have an Excel file that has a bunch of VBA and macro code in it.  When I open the file in Excel I can choose not to 'enable' them - so the values in the fields all stay as they were during the last save.  I need to manipulate the values as they were last saved - so I don't want the macros (which look at the current date and update values accordingly) to run.  When I open it via our dot net code:   Dim oxlRep As Excel.Application Dim oWBRep As Excel.Workbook Dim oSheetRep As Excel.Worksheet Dim oRngRep As Excel.Range oxlRep.Open(path)   the vb code runs - throwing off the values.  I've been looking for a way to open it without macros, or in 'secure' mode where the macros aren't run.  If I simply double click the file and don't choose to enable macros the values are all there as I want them.    Usually we run this code within the month that the files are created, so we haven't seen this problem in the 3 or 4 years that it has been working.  Now I need to go back to some of the old files and run some archival code...    Anyone have a suggestion? I asked  that turned out to have multiple correct answers. How do I pick between them to determine which to accept?  All of the answers were different. They were not the same answer given multiple times. Coloring lines inside textarea