In our discourse FLT is Fermat's Last Theorem. I am unaware of any theorems or conjectures that begin assuming FLT is true, or otherwise use FLT as a starting point or tool. The small amount of literature review I've done on this question reveals nothing.  My question is: Where can I find a work requiring FLT, or some useful implication of FLT? Even an implication of a polynomial inequality, that may not be FLT, may be a good answer to this question, as I'll likely try to use it to find something regarding FLT.  The following is not acceptable as an answer to this question:  $$ (a^{x_{1}}_{1} + b^{x_{1}}_{1} - c^{x_{1}}_{1}) \ldots (a^{x_{n}}_{n} + b^{x_{n}}_{n} - c^{x_{n}}_{n}) \not= 0 : x_{i} &gt; 2 $$  and it's expansions imply (something trivial)  Another acceptable answer to this question would be a proof requiring FLT to be false.  Thanks and please let me know if I can ask this question in a way more fitting math.se (new user).
I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number Theory leading to the development of new methods which can be themselves applied not only for the proof of FLT (like financial problems motivated somehow development of stochastic analysis). I am interested - if there are applications or implications of FLT itself?  More precisely: if the fact "for each $n\geq3$ there are no integer solutions of $a^n+b^n=c^n$" leads to solutions of problems which are not in the field of Number Theory?   I would specify that I wonder about some problems which are already formulated: since FLT is known for more then 300 years I am pretty sure that there were formulated hypothesis which follow from FLT directly (if there are such hypothesis not in the field of Number Theory).
Feedback for The Loop - May 2020 May’s “The Loop” blog post was just posted, you can read it . A popular response last month to our post about how we receive, prioritize and implement feedback was a question around how this framework gets utilized in the practical sense. We thought that Dark Mode was a great feature to dig down into as it was so widely requested and anticipated.   We hope you enjoy this month’s post, please leave any questions or comments you have in this thread and we will respond ASAP. I found native people prefer to use passive voice in a sentence. For example:  Tonight's moon can be seen from anywhere worldwide. We have reason to believe that progress can be made.  Be honest, it's difficult for an English learner to understand. Why not phrase it this way:  We can see the moon from anywhere worldwide.  In which cases do people prefer to use passive voice rather than active voice? Is there a way to make rigid body interact with softbody physics? Unable to locate package Installing PHP 5.3 in Ubuntu 14.04 I've just installed Ubuntu 14.04 and am trying to install php 5.3. But when I run   apt-get install php5    it's installing php 5.5.9-1ubuntu4. I need to use php 5.3 in the system because the application I'm working on is built with php 5.3.   How can I achieve this? Will this schematic work as a four-bit shift register? &ndash; Schematic created using   Will this work as a four-bit shift register? (And by extension a shift register of any size?) Each vertical PMOS / NMOS transistor pair acts as a multiplexer, and the PMOS transistor to its left amplifies the voltage back up to close to V_high. I'd be interested in hearing people's feedback on this. Right now I have only three transistors (and a resistor) per bit. Can I really get it that small a ratio, or are there reasons why I need more transistors per bit?  It's probably worth noting that I don't have master/slave here, so I can only assert (Shift) long enough to move one bit toward the right and no more.  Furthermore, I'm not trying to say that I need a resistor with 100 ohms. That appeared to be the only resistor I could use to make this schematic! What I want is a resistor that has significantly less resistance than the PMOS it's next to when it's at high impedance, but enough resistance to prevent a short circuit from V_high to ground when that transistor is conducting.  Okay, now I've changed each of the BJT's to MOSFET's, and reduced the resistance of each resistor from 100 ohms to 10 ohms. I'm still going on the assumption that (shift) will only be asserted long enough for the gate delays of M4 and M5, and then of M6, so the value of (dataIn) will only make it to M6's output, and won't move to M8 until the next time (shift) is asserted. Is this enough to get this circuit behaving like a four-bit shift register? So, I just finished off the first part of an important task for school and today when I was going to continue, the document with my notes were gone. The only fragment I've got left of my file is the back up file, which doesn't show in the folder where I saved the document, but I can search for it.  Is there anyway which allows me to open the back up file, save it as a text file and continue my task? I don't really have the time for running restoring tools, which never has helped me in a satisfactory way anyway.  I wrote the document in gedit and my version of Ubuntu is 12.10. Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? Let $X$ be obtained by removing finitely many lines from $\mathbb{R}^3$. How do I prove $X$ is connected? Let $X$ be obtained by removing finitely many lines from $\mathbb{R}^3$. How do I prove $X$ is connected?  I think the way to go about this is to show $X$ is path-connected but I'm not quite sure. Why the definition of limit have a strict inequality rather than not? My question is why the definition of limit of sequence is for $\epsilon &gt;0$, there exist $N$ st for $n\ge N$ that $|x_n-L|&lt;\epsilon$ instead of $|x_n-L|\le \epsilon$ I'm stuck in a teleporting loop. What can I do? How to use GeoPandas buffer function to get buffer zones in kilometers? Does blocking an attack with the Shield spell still trigger Armor of Agathys? I am contemplating a Oath of Conquest Paladin / Sorcerer multiclass character, with a 1 level dip in hexblade warlock. This character will have access to the Armor of Agathys and Shield spells, which potentially allows for a pretty nice combo. The Shield spell states (PHB 275):  Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell... Until the start of your next turn, you have a +5 bonus to AC, including against the triggering attack,  It looks like you are indeed hit by an attack, but since you have a bonus to AC, you can potentially force that attack to miss instead. Armor of Agathys states (PHB 215),  If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.  If I cast Shield and cause an attack that &quot;hit&quot; me to miss, does Armor of Agathys still deal damage? I am leaning toward yes, though this combo seems a little too strong--Shield is already a strong spell, and this character will eventually have 9 1st level spell slots and a 7th level spell slot. Not only can they freely block attacks with Shield, but they can also deal up to 35 (5d10) damage to the attacker without an attack roll or saving throw. While a Warlock would have to spend both of their spell slots to do this once, my character would be able to do this 4 times with spell slots to spare. Some notes:    ask about reducing the damage from a confirmed hit to zero. This question is asking about what happens when the Shield spell causes the attack to miss entirely. A number of online resources, such as the Roll20 5th edition compendium, seem to be missing the correct casting time for the Shield spell. Answerers should make sure they're looking at the correct version from the PHB. My server had been infected and the attacker spread a bunch of .5randomletters folders around my file system, with malicious code inside of them. I want to search for all the folders that begin with a dot, review them and then remove them.  How can I do that? When will my phone get the Android 2.3 update (Gingerbread)? How do I iterate over a range of numbers in Bash when the range is given by a variable?  I know I can do this (called "sequence expression" in the Bash ):   for i in {1..5}; do echo $i; done   Which gives:     1    2    3    4    5   Yet, how can I replace either of the range endpoints with a variable? This doesn't work:  END=5 for i in {1..$END}; do echo $i; done   Which prints:     {1..5} People often use sans, the French word for ‘without’ in English.   What I don’t get, is when should that switch be made. When should I say sans when I really want to say without. Are there any general rules regarding usage, or is it kind of a person-to-person basis/feeling of flowerness. Gauge fields and strings: Loop equations How do I sort a vector of pairs based on the second element of the pair? If I have a vector of pairs: std::vector&lt;std::pair&lt;int, int&gt; &gt; vec;  Is there and easy way to sort the list in increasing order based on the second element of the pair? I know I can write a little function object that will do the work, but is there a way to use existing parts of the STL and std::less to do the work directly? EDIT:  I understand that I can write a separate function or class to pass to the third argument to sort.  The question is whether or not I can build it out of standard stuff.  I'd really something that looks like: std::sort(vec.begin(), vec.end(), std::something_magic&lt;int, int, std::less&gt;()); Read MIME data using PHP