Can we argue that the square of every odd integer is of the form $8k+1$ using non inductive proof?
How can we prove that every odd perfect square is congruent to $1$ modulo $8$?
Each number in a subset $S\subseteq \{1,\ldots,2n\}$ does not divide another one. Then $\max |S|$? This problem comes from a seemingly innocuous question from a professor during a lesson for a Math Olympiad course. [A part of this question is really a classic of number theory/combinatorics]  Problem: Let $S$ be a a non-empty subset of $\{1,2,\ldots,2n\}$. It is clear that if $|S|$ is "enough large" then, independently from the choice of $S$, there are for sure some distinct $a,b \in S$ such that $a$ divides $b$.  Then, what is the maximal $k$ such that, for some $S$ with cardinality $k$, if $a,b$ are distinct members of $S$ then $a$ does not divide $b$?  [Ps. I know the solution] What is the most CPU efficient way to ensure that NumLock is on? Situation: I have a script watching over NumLock status, well not really watching, but turning it On every 1 second. The script is running in the background.  Reasoning: I often accidentally turn off NumLock. And I have no indicator of NumLock status on the keyboard.  OS, DE, DM, WM, xmodmap:  :  $ lsb_release -a  No LSB modules are available. Distributor ID: LinuxMint Description:    Linux Mint 18 Sarah Release:    18 Codename:   sarah   :  echo $DESKTOP_SESSION  cinnamon   :  cat /etc/X11/default-display-manager  /usr/sbin/mdm   :  wmctrl -m | head -n 1  Name: Mutter (Muffin)   :  xmodmap -pm  xmodmap:  up to 4 keys per modifier, (keycodes in parentheses):  shift       Shift_L (0x32),  Shift_R (0x3e) lock        Caps_Lock (0x42) control     Control_L (0x25),  Control_R (0x69) mod1        Alt_L (0x40),  Alt_R (0x6c),  Meta_L (0xcd) mod2        Num_Lock (0x4d) mod3       mod4        Super_L (0x85),  Super_R (0x86),  Super_L (0xce),  Hyper_L (0xcf) mod5        ISO_Level3_Shift (0x5c),  Mode_switch (0xcb)   My original Bash script follows:  #!/bin/bash  while true do    numlockx on    sleep 1s  done   As you can see, the script does not care about current status of NumLock. It just keeps turning it on.  Goal: I would like to make the script at least somewhat CPU efficient.  Question: What is the most CPU efficient way to ensure that NumLock is On in Linux (Mint 18)? Splicing a list of arguments into a function with Sequence Does the VBA "And" operator evaluate the second argument when the first is false? How can I add a Login link to the "Main navigation" menu?  When a user clicks on the link, it should take them to the /user/login page. Once they login, the link should change to say "Logout" instead of "Login".  Is there any way to add such a link? Verb/Subject Agreement Make/Makes So I was reading this:  and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any.  So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative?   EDIT Ok, I changed my question a little bit:  Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.)  Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&amp;Lifshitz's mechanics book only consider this kind of change in the Lagrangian? Parsing DateTime on Localized Systems It would seem that far-away stars are at such a distance that I should be able to take a step to the side and not have the star's photons hit my eye. How do stars release so many photons to fill in such great angular distances? Intuition behind the definition of a measurable set A linking verb is a verb which connects a subject to its predicate   without expressing an action. A linking verb is used to re-identify or   describe its subject.      Ex:      Alan is a beast.      Alan seems drunk.   But I would say that some linking verbs can be used with "continuous tenses" &amp; other linking verbs can not, but I am not sure.      (linking verb) to experience a particular feeling or emotion      Ex: I’m feeling a little better today.      Do you still feel hungry?   So, feel in this meaning can be used with continuous tenses  But I would think this sentence "Do you still feel hungry?" does not make much sense because it uses "simple present tense" which expresses something that happens regularly. Noone would feel hungry regularly.  So, that sentence "you feel hungry" would mean "you always feel hungry".      (linking verb) (not used in the progressive tenses): to have a   particular physical quality which you become aware of by touching         adj.         The water feels warm.      Its skin feels really smooth.    says "Speakers sometimes use the Simple Present to express the idea that an action is happening or is not happening now. This can only be done with Non-Continuous Verbs and certain Mixed Verbs."  so, "The water feels warm." could mean "The water is always warm" (simple present expressing things that happen regularly - more or less permanent action)  But "The water feels warm." could mean "The water is warm now" (simple present with non-continuous verb expressing things that is happening now - a temporary action)     : (linking verb) to seem; to appear         adj.         to look pale/happy/tired      That book looks interesting.   In dictionary,      (linking verb) (not used in the progressive tenses): to   give the impression of being or doing something         adj.         She didn't appear surprised at all.       (to somebody) (to be) something (not used in the progressive tenses): to give the impression of being or doing something (synonym:   appear)         adj.         You seem happy.   So, we cannot say "you are seeming happy" or "you are appearing happy".  But I am not sure I can say "You are looking good" (sounds right to my ears) or "That book is looking interesting." (doesn't sound right to my ears)  But we can say "The future’s looking good." ()  So, some verbs has the same meaning but one can be used with "continuous" &amp; the other can not, right?  Ex: "you are looking happy", but "you seem happy". I have a Dell N5110 laptop with integrated Nvidia GeForce GT 525M and an Intel HD Graphics 3000 card; My problem is the default card is the Intel one, and it provides a very bad performance with Windows 8. So can I force to use the Nvidia one all the time including "Desktop Windows Manager" ? How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$  where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio? Robot tank run amok, its trainer trapped in cave on the Moon Another dim memory from my sci-fi reading binges in the 1980s... this one almost definitely in one of the big hardback collections full of late Golden Age short stories.  The story is told from the perspective of a robot (maybe cyborg?) tank that has stopped responding to control signals and is aggressively "defending" some portion of the Moon, to which it was previously assigned. Maybe a war had ended, or maybe it became confused about an exercise that it thought was real?  There was a person trapped in a cave trying to communicate with the tank. The tank mostly ignores him, so the main purpose of what he says is exposition for the reader. I think this person was an expert on this type of robot/cyborg sent to "talk it down", but the tank had attacked, and this guy was the only survivor. The expert was badly wounded -- I think he mentions his suit had to amputate his leg to seal a hole -- or maybe flood it with water which would freeze to seal the hole but mean that his leg had to be amputated later. He keeps trying to convince the tank to stand down, at one point saying that he knew it "back when it was just a two-wheeler an autocyber in boot camp" or something along that line. (Edit: I must have been reading too fast and/or distracted back then, since per the quote below I seem to have misread "motorcycle" for "autocyber"!)  One detail of the attack stands out -- the tank was low on ammunition, I think, so it used a kind of bomb with an electric/magnetic "catapult" (aka "magnapult") to hit the expert's transport. The tank was proud of itself for figuring out a way to hit it far out of the normal range by 1) using its own top speed to add velocity to the projectile, and 2) building up the catapult's energy past its normal operating range, which caused it pain but made the bomb go much further. It catches the transport by surprise, since they thought they were safely out of range of the armaments it had left.  The premise sounds a lot like a Keith Laumer "Bolo" story, but I'm fairly sure that it's not actually one of them. It had a somewhat grimmer feel to it. I could be wrong, though, since it's been 30 years. Sum of nilpotent and element is a unit in ring? I can't tell where I went wrong, am still a beginner.    Let N be a nilpotent in a commutive ring , let X be any element.   I'll will be showing X + N is a unit .   Assume Y(X+N) =1   Then YX(N^n-1) =  N^n-1   Multiplying both side by  X+N   We get X(N^n-1) =  X(N^n-1)  Thus nilpotent summed with any element gives unit. How do I stop an effect in jQuery I have a page that uses   $(id).show("highlight", {}, 2000);   to highlight an element when I start a ajax request, that might fail so that I want to use something like  $(id).show("highlight", {color: "#FF0000"}, 2000);   in the error handler. The problem is that if the first highlight haven't finished, the second is placed in a queue and wont run until the first is ready. Hence the question: Can I somehow stop the first effect? Trying to hide spam/offensive deletions on user profiles announces them When are stopovers permitted on Japan Railways (JR) trains? Regarding : in general, what are the rules governing stopovers for JR trains if one does not use a rail pass? A stopover means you have a ticket from stations A to B (say, Tokyo to Hiroshima) and you disembark and exit the ticket gates at an intermediate station (say, Kyoto).  (Spoiler alert: it's complicated.) For some reason all of a sudden when I update cells in Excel, my equations are not being updated.  For example, I have a simple equation =SUM(C4:C16).  If I change any of the cells in C4-C16, the answer doesn't get updated...  I don't know what happened I've never seen this fail before.  Any ideas? In a technical environment, what is the most suitable sentence to use when answering to someone about a problem that they had and we solved it for them:   The problem is solved The problem has been solved