Is there a positive integer $N$ such that the equation $x^2 + y^2 = N$ has at least $2005$ solutions in non-negative integers $x$ and $y$? If so, prove it. If not, prove it!
I'm looking for some help disproving an answer provided on a  I posted about computing the number of double square combinations for a given integer.    The original question is from the Facebook Hacker Cup       Source: Facebook Hacker Cup   Qualification Round 2011      A double-square number is an integer $X$   which can be expressed as the sum of   two perfect squares. For example, 10   is a double-square because $10 = 3^2 + 1^2$.   Given $X$, how can we determine the   number of ways in which it can be   written as the sum of two squares? For   example, $10$ can only be written as $3^2 + 1^2$   (we don't count $1^2 + 3^2$ as being different). On the other hand,   $25$ can be written as $5^2 + 0^2$ or as   $4^2 + 3^2$.      You need to solve this problem for $0 \leq X \leq 2,147,483,647$.      Examples:      $10 \Rightarrow 1$      $25 \Rightarrow 2$      $3 \Rightarrow 0$      $0 \Rightarrow 1$      $1 \Rightarrow 1$   In response to my original question about optimizing this for F#, I got the following response which I'm unable to confirm solves the given problem correctly.       Source:       Again, the number of integer solutions   of $x^2 + y^2 = k$ is four times the   number of prime divisors of $k$ which   are equal to $1 \bmod 4$.      Knowing this, writing a program which   gives the number of solutions is easy:   compute prime numbers up to $46341$ once   and for all.      Given $k$, compute the prime divisors of   $k$ by using the above list (test up to   $\sqrt{k}$). Count the ones which are   equal to $1 \bmod 4$, and sum. Multiply   answer by $4$.   When I go through this algorithm for $25$, I get $8$ which is not correct.          For each prime factor (pf) of $25$ ($5$, $5$ are prime factors of $25$)         if pf % $4$ = $1$ (true for both $5$'s), add $1$ to count      return $4$ * count (count would be $2$ here).          So for $25$, this would be $8$
I'm trying to set up a way for me to just type newjournal, and a new journal entry named year-month-date-time.txt is created in a year subdirectory in ~/documents/journals.   I've got it set up as an alias using the following:  alias newjournal='nano /home/username/documents/journals/'$(date +%Y)'/'$(date +%F-%k%M).txt''   So the year subfolder resolves correctly, and seemingly so does the first half of the date command. Tried appending the alias with sudo, didn't help.  Issue: For some reason, when I do this command the time value (%k) resolves to a few minutes ago... or maybe the last time nano ran this filename, maybe it's called an old buffer?   When I logoff it makes the correct file, then I save, wait a few minutes and try again, and it edits the old file instead of making a new one. Even if I remove the old file, it creates a new file with the old filename (incorrect time).  Here's a screenshot of what I'm trying to explain. Here I create a newjournal, save in nano within a second or two of executing the command, print the correct date/time, list the file that was created. Note the incorrect filename time (1249, as apposed to 13:11):    Anyways, it might be something simple, but I've tried different methods in the alias, such as putting double quotes around the whole thing and escaping properly, single quoting, escaping and spacing, etc.  Thanks for the read/help. Ubuntu 16.04 server.  EDIT: Here's the final alias that worked:  alias newjournal='nano /home/user/journals/"$(date +%Y)"/"$(date +%F-%H%M).txt"'   Needed to double quote the date commands inside the overall single-quoted alias. We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. Here is a problem which i have not been able to do for quite sometime.   For $n \geq 2$, show that $n \nmid 2^{n}-1$.   I have thought of proving this in two ways: One by using induction which didn't actually work. Next by Fermat's little theorem we have $2^{p-1} \equiv 1 \ (\text{mod} \ p)$, which actually says that for $p \mid 2^{p}-2$. But i couldn't proceed more than this. Any ideas by which i can actually solve the problem? Wolfram alpha says there are no solutions. Is this because of the following?  $$\tan(z)=i \Rightarrow \frac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})}=i \Rightarrow \frac{e^{iz}-e^{-iz}}{(e^{iz}+e^{-iz})}=-1$$ Let $x=e^{iz}$. Then,  $$\frac{x-1/x}{x+1/x}=-1 \Rightarrow x-1/x=-x-1/x \Rightarrow 2x=0$$  Then, $$x=e^{iz} \Rightarrow \ln(0)=iz$$ but $\ln(0)$ does not exist so we have no solutions? How do I find out which DOM element has the focus? I would like to find out, in JavaScript, which element currently has focus. I've been looking through the DOM and haven't found what I need, yet. Is there a way to do this, and how?  The reason I was looking for this:  I'm trying to make keys like the arrows and enter navigate through a table of input elements. Tab works now, but enter, and arrows do not by default it seems. I've got the key handling part set up but now I need to figure out how to move the focus over in the event handling functions. Proof that the real numbers are countable: Help with why this is wrong I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers.  First, take the numbers between zero and one, exclusive.  The following sequence of real numbers is suggested so that we have bijection.    0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.10, 0.11, ... , 0.99, 0.001, 0.002, ... , 0.999, 0.0001, etc.  Obviously, this includes repeats, but this set is countable.  Therefore, the set of all numbers between zero and one is a subset of the above countable set, and is thus countable.  Then we simply extend this to all real numbers and all the whole numbers themselves, and since the real numbers, as demonstrated above, between any two whole numbers is countable, the real numbers are the union of countably many countable sets, and thus the real numbers are countable.    Please help me with this.  I understand the diagonalization argument by Cantor, but I am curious specifically about this proof which I thought of and its strengths and flaws.  Thanks. Not getting edit's +2 rep after reaching 2k rep After reaching 2k rep , I'm not receiving  +2 rep per edit. Why? Marriage Tax Benefits How to tell the difference between linear and non-linear regression models? I was reading the following link on non linear regression . My understanding from reading the first section "Nonlinear Regression vs. Linear Regression" was that the equation below is actually a linear regression, is that correct? If so why?  $$y = b_1x^3 + b_2x^2 + b_3x + c$$  Am I also to understand that in non linear regression multicollinearity isn't an issue? I know that multicollinearity can be an issue in linear regression so surely if the model above is in fact a linear regression there would be multicollinearity? Polynomial in $n$ variables Let $f$ be polynomial in $n$ variables. If we have $f(a_1,\ldots,a_n) = 0, a_i \in S_i$ and $S_1,\ldots,S_n$ are infinite subsets of a field $k$, how can I show that $f=0$? Double vs. BigDecimal? Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ This is a response to the following exchange at    [Bill constructs a $\aleph_0$ linearly independent set in $\Bbb R$]  Can you extend the idea of this proof to get the right dimension? Currently, you only have a countably infinite independent set, but the dimension is size continuum. –   I believe the answer to this question is no, for the following reason. Suppose we have a spanning set for $\Bbb R$ over $\Bbb Q$, of cardinality $\frak n$. Then by definition we have the cardinality equation ${\frak c}\le^*[{\frak n}]^{&lt;\omega}\cdot \aleph_0$ (where $[{\frak n}]^{&lt;\omega}$ is the cardinality of finite subsets, and $\frak m\le^*n$ means there is a surjection from $\frak n$ to $\frak m$). If $\frak n$ is finite, then we get ${\frak c}\le^*\aleph_0$ which is false, so $\frak n$ is infinite (but not necessarily Dedekind-infinite). If we assume the axiom of choice, then the equation reduces to $\frak c\le n$, hence every Hamel basis is size continuum. But without AC I don't know any way to simplify the original equation to get anything stronger than &quot;$\frak n$ is infinite&quot;. This is at best heuristic evidence for the original question, since it is considering a different problem (spanning sets instead of linearly independent sets). How could one show that there are no ZF-constructible $\Bbb Q$-linearly independent subsets of $\Bbb R$ of cardinality continuum? Why are springs shaped the way they are? Why are springs coiled the way they are? Why not some other shape? Is the shape due to its elasticity or something? Blackboard bold variants for Greek letters It is common (or it used to be) to use a blackboard-bold variant of \mu to denote roots of unity in some mathematical papers.  However, despite searching (several times, periodically over several years) I have never been able to locate any package or technique that makes this possible, other than using Metafont or possibly hunting through the font definitions to see if it is something I can create a variant of (along the lines of the suggestions in Section 8 of the Comprehensive LaTeX symbol list).  Thus far, I haven't been sufficiently motivated to learn how to do any of this (although it probably would have resulted in a net time savings), and I feel like I cannot be the only person who would have tried.  Is anyone aware of an existing way to produce this character? href to an image label - how to jump to the image instead of the caption below the image? I'm not a resident of USA. But I work on a cruise ship which required me to have C-1/D visa. I am leaving USA and traveling to Mexico but I will have a connecting flight somewhere in another US airport with 1-2 hours waiting for connecting flight to Mexico.   Do I need B1/B2 visa for that? Or my transit visa C-1/D will be enough as long as I am leaving USA? Working with large numbers in PHP Problems with character input using scanf() This is a summary of the excerpt:  Given that we have:  *$10^9$ people  *each person has a chance of 0.01 of going to a hotel  *we have $10^5$ hotels, each hotel holds 100 people  *Thus, $P(2 \text{ persons meet on the same hotel in any given day}) = 0.01 \cdot 0.01 \cdot 10^{-5}$  *and $P(2\text{ persons on the same hotel in $2$ day}) = P(2 \text{ persons on the same hotel in any given days})^2 = 10^{-18}$  the book "Mining of Massive Dataset" said that the expected number of pairs who stay in the same hotel for 2 days is:  $P(2 \text{ persons on the same hotel in $2$ days}) \cdot ( \text{number of pairs of people} ) \cdot (\text{number of pairs of days})$  this is the part where I got stuck. Can't understand how it come to this derivation. Can anybody explain it in more elaborate detail.  You can find the content on page 6 of this link:  (it is freely available) I know that the movies were written to be a stand alone universe on their own, but what key elements were missing in the movies that were vital in the books that viewers are missing out on?