Let $a \in \Bbb Z$ be such that $a = b^2 + c^2,$ where $b,c \in \Bbb Z \setminus \{0\}.$ Then $a$ cannot be written as $:$   $(1)$ $p d^2,$ where $d \in \Bbb Z$ and $p$ is a prime with $p \equiv 1\ \left (\text {mod}\ 4 \right ).$   $(2)$ $p q d^2,$ where $d \in \Bbb Z$ and $p,q$ are distinct primes with $p,q \equiv 3\ \left (\text {mod}\ 4 \right ).$  $(1)$ is false because $2^2 + 1^2 = 5 = 5 \cdot 1^2,$ where $d = 1 \in \Bbb Z$ and $5$ is a prime with $5 \equiv 1\ \left (\text {mod}\ 4 \right ).$ How do I prove or disprove the other option? Any help in this regard will be appreciated. Thanks for your time.
Let $a\in \mathbb{Z}$ be such that $a=b^2+c^2$ where, $b,c \in \mathbb{Z}-\{0\}$. Then $a$ cannot be written as    $pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 1~(mod~4)$ $pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 3~(mod~4)$ $pqd^2$ where $d \in \mathbb{Z}$ and $p,q$ are primes with $p \equiv 1~(mod~4)$ and $q \equiv 3~(mod~4)$ $pqd^2$ where $d \in \mathbb{Z}$ and $p,q$ are primes with $p,q \equiv 3~(mod~4)$   The similar question with some part missing is asked , but I want to discuss additional options too.  Here, I got 1) is false Since $13=3^2+2^2$ For other options What should I do
How do I get the icon associated with a file type? I have a form that displays file information in a TabControl, and I'd like the pages to have the file's icon in their tab.  How do I get the icon associated with a file type?  I'd prefer solutions that don't involve looking things up in the registry, but if that's the only way then so be it. Why do modern mountain bike forks no longer use boots on fork stanchions? Why do modern mountain bike forks no longer use boots on fork stanchions? Older mountain bikes used to have them on the forks, and lefty and headshocks still do. Why have these fallen out of fashion and would putting aftermarket fork boots on your stanchions be productive or not? How can I resize NTFS partition in GParted? In a Cycles scene I have an equirectangular image serving as an Environment Texture background (set in the World settings).  I have a glossy object being partially lit by the Environment Texture, as well as getting reflections from it.  Is there a way I can exclude the Environment Texture from affecting the glossy object completely? I want the image as background wallpaper only. The glossy object should be affected by other objects and lights within the scene: everything except the Environment Texture. SEO Removal Time? I was searching about removing the a domain/site being listed in Web Search Results (like, Google Search) and i found solution, .  I believe it will be the solution but what i still need to know is:   How long will it take to bring down my Website being listed from Search Engine Results? How to install (old) packages for Ubuntu 9.04? Based on some excellent feedback by Mark here (), today I was able to use the vmware converter to clone my production server for a staging server.  However the nic won't come up no matter what I do. I attempted to inistall vmware tools, as I suspect that the fact that it is not installed might prevent the nic from working. (I have the nic set as a vmxnet3 card in the vm settings). The install failed because there were several dependencies missing as well as the Linux headers. Given that Ubuntu 9.04 has been EOL'd, the packages I need to install to get the vmware tools to install are no longer available. I doubt the ubuntu 9.04 install CD has the packages on it.   What are my options? I'd rather not upgrade the version of Ubuntu yet, as the point of the vm right now is to maintain parity with the production server. Might I have better luck resetting the driver to use vmxnet2 instead of the vmxnet3?  Thanks in advance!  Chris What are metaclasses in Python? In Python, what are metaclasses and what do we use them for? When you call the object.__repr__() method in Python you get something like this back:    &lt;__main__.Test object at 0x2aba1c0cf890&gt;     Is there any way to get a hold of the memory address if you overload __repr__(), other then calling super(Class, obj).__repr__() and regexing it out? How can I find a mentor to critique my work? One thing that made me curious while watching a football game on TV was the numbers that appear on the ground. For instance, there's a "1st &amp; 10" beside one team, but there was a "3rd &amp; 6" a few minutes prior (and I think it was beside the other team). What do these numbers mean? When does the 'Lost City' become Atlantis in Stargate In Stargate SG-1 they are searching for the Lost City in season 8, however in the Stargate Atlantis pilot it is referred to as Atlantis. When is the name found? Or the connection between the Lost City and Atlantis made? Identifying pin numbers from PCB CHECK constraint in MySQL is not working Non-Exchange address in Exchange GAL We just made the switch to Google Apps for part of our domain. We disabled the accounts that are now on Google Apps and mail routes correctly. Our problem is that now we have no way of seeing those users in the Global Address List.  So we have these users in our Active Directory, they have no mailbox on Exchange, and we want to be able to see/search for their addresses in the GAL. Is there any way to do this?  This is Exchange 2007, if that matters. Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$ How do you determine the size of a file in C? How can I figure out the size of a file, in bytes?  #include &lt;stdio.h&gt;  unsigned int fsize(char* file){   //what goes here? } We've had a number of instances over on Stack Overflow involving users who get to the reputation level where they can comment, then spam comments across the site. Due to the lack of community review around comments, these spammed comments can remain on the site for months before someone notices and flags one. Moderators then have to go and manually remove each of these comments. In the last few months, I've personally had to deal with two users who spammed their website across 200+ questions each.  This isn't just limited to people spamming comments with a URL in them. We've had a number of users spray the same nonsense comment or insult across tons of posts and have no one seem to notice for months.  What I'd like is some kind of system-generated notification when a user has posted a barrage of copied-and-pasted comments. This could possibly be triggered when a user has posted a set number of identical comments in a row (4 of them, maybe) or maybe even if they've posted the same comment multiple times. We receive similar system-generated flags when users post duplicate answers, edit many of their posts in a row, or delete many of their own posts at a time.  This system-generated flag should ignore comments generated by review (which have the same text in them). I'm thinking that it shouldn't ignore comments generated by custom off-topic reasons, because someone using the exact same close reason over and over again might be something we want to look into.  The flag itself won't do anything to a user, and would only appear to moderators. I imagine there will be a number of false positives, but it should be easy for us to check these and identify the well-intentioned stock comments from the problematic ones.   We currently have no means of detecting comment spam, so even if this didn't have a great hit rate it would still be beneficial on the balance. Running sample queries for this seems to indicate that it wouldn't have that much noise.  This would allow us to catch comment spam early, and prevent it from living on the site or developing into a more serious problem. Attributive nouns vs. of-genitive Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? How do you write something that was discussed in the past, while the issue still exists in the present?        The discussion resulted in the committee members highlighting crucial areas that need to be addressed.   The discussion resulted in the committee members highlighting crucial areas that needed to be addressed.      ......................     1'. The discussion resulted in the committee members highlighting crucial areas that need to be addressed if one is to achieve the desired outcome.      2'. The discussion resulted in the committee members highlighting   crucial areas that needed to be addressed if one were to   achieve the desired outcome.