Frobenius coin problem using Mathematical Induction.
Proof that every number ≥ $8$ can be represented by a sum of fives and threes.
Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b&gt;0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any hint will be appreciated. Reading a huge .csv file How do I technobabble without sounding like technobabble? Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks I am running out of storage, I usually have bought a lot of my games on disk but if I buy a game on the store then delete it, and want to redownload it do I have to buy it again Best File Management Practices when you have thousands of Files Show that every large integer has a large prime-power factor  That is, if $P(n)$ designates the largest number $p^a$ which divides $n$, then $\lim_{n\to\infty}P(n)=\infty.$ Aligning Vertices Diagonally Is there a way to remove an overlap from multiple polygons to make one final polygon?..programmatically, I am using the GeoTools or/and OpenGIS/JTS.. any ideas to put me on the right path...  example initial:  final: How to change a default application in Ice Cream Sandwich? I used the ADWLauncher EX by default while having Gingerbread 2.3.6.  Today I have manually updated my Nexus S to Ice Cream Sandwich 4.0.3, and I wanted to give the new default launcher a shot, but I cannot find anymore where to select / forget the default application for an action.  I've turned the settings of my phone inside out, googled and crawled the forums, but without success. If anybody can tell me how to change a default application in ICS, I'd be grateful!      Yeah, I never thought I'd want to see that Complete action using... window again! :D What is safe to exclude for a full system backup? Last activity on question shows modified by user who did not ask or answer .       The  has no sign of the user sharth The  also does not have any sign of related activity   I would like to know why the user's name is associated with the question. At least why their display name shows up as having modified the question timeline activity.  Possible reason that I can think of is that, .  Is this the reason?  Is there a way (for normal users), to look for such details, other than looking at the user activity and the question's revision history. Selecting all objects that have been specified in Modifiers Could anyone help in making a script for selecting all objects that are specified in the different modifiers that an object has stacked? For example if my object has a Boolean modifier and an Array modifier, which both refer to an empty or another object (in the Array modifier there is for example an Object offset you can specify). Would it be possible to select all those object with one click? Thanks. How can I get Google to stop thinking I'm a Bot? So, I'm getting the occasional CAPTCHA message, which is irritating but fine, but occasionally Google just blocks my network wholesale. I've submitted automated requests to unblock it, but I have no idea who I can email directly to clarify the issue.  I've run all manner of anti-virus, parsed my router's logs, combed through Fiddler to see if any errant packets were slipping by. Nothing. The only Google searches that happen are made by me through the web-browser.  Who or what do I have to do to get Google to stop falsely flagging my network as a spam-bot? Generate a log of all the installs on a system (whether apt-get/yum/pip/npm/...) I have a many-to many relationship between three objects. LGFProducts, PartsinProducts, and Parts. PartsinProducts is the junction object. I want to be able to get data like the names of the parts that are related to the selected LGFProduct.   I understand that parent-child relationship can only traverse one level down while grandchild-grandpa can travel 5 levels at most. What is the parent-child or child-parent in a many-to-many relationship?  I assumed that I should be traversing upwards from Parts to the junction object partsinproduct and then LGFProducts? But my friend who is not a Salesforce expert but has knowledge in CS said that I should do a parent-child query (of LGFProducts to partsinproduct) to child-parent relationship query (PartsinProduct to Parts). Or perhaps both of these options are wrong and another is appropriate.   Also, I am having a little trouble traversing for custom objects. I understand that I need to append the letter r instead of c but get confused. For example, this is my attempt at a three level upward traversal that fails.   SELECT Id, Name, Parts__r.PartsinProducts__r,LGFProducts__r.Name FROM Parts__c   Also if three levels of upward traversal are needed, could one provide an example with custom relationships? As seen above, I am having difficulty successfully querying a three level upward traversal statement for custom objects.  This is my attempt at the parent-child relationship query of LGFProduct to PartsinProducts:  SELECT Name,   (     SELECT Name     FROM PartsinProduct__r   ) FROM LGFProduct__c    I am then given this error: Is there an idiom for people who boast too much? I am looking for idioms or informal/slang/colloquial expression for some people that make you think that they are able of building a skyscraper, constructing a spaceship, playing the piano better than Mozart or something very fancy that requires remarkable skills, but when it comes to practice they prove to be completely inadequate. Does Windows 7 allow auto-login with a stored password, like TweakUI did? The TweakUI powertoy for Windows XP had an option to auto-login. It stored your password and did the login for you. This was useful, since you kept password protection, but had login programs loaded when you first booted.  When you switched users or hibernated, it would still ask for a password, which is the desired behavior.  Is there a way of doing this in Windows 7? How to calculate $\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$ I came across this strange limit whilst showing convergence of a series: $$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$$ How can I calculate this limit? Scorched3d worked then stopped. Unistalled and reinstalled did not help. Running Ubuntu 16.04.04LTS. I think it was after installing new release.  I think the problem is the Ubuntu 16.04.4LTS. The newer version has no Scorched3d in the Ubuntu Software Center. I have 3 systems. After updating to new version. Scorched3D will not install and work after updating.  ron@RonsComputer:~$ sudo objdump -p $(type -p scorched3d) | grep NEEDED  NEEDED libwx_gtk2u_adv-3.0.so.0  NEEDED libwx_gtk2u_core-3.0.so.0 NEEDED libwx_baseu-3.0.so.0 NEEDED libfreetype.so.6 NEEDED libz.so.1 NEEDED libSDL-1.2.so.0 NEEDED libSDL_net-1.2.so.0 NEEDED libexpat.so.1  NEEDED libstdc++.so.6 NEEDED libm.so.6 NEEDED libgcc_s.so.1 NEEDED libpthread.so.0 NEEDED libc.so.6   ron@ron-desktop ~ $ objdump -p $(type -p scorched3d) | grep NEEDED | awk '{print $2}' | xargs -n 1 dpkg -S  libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0.2.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6.12.1 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6.12.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1.2.8 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1.2.8 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0.8.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0.8.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1.6.0 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1.6.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1 libstdc++6:amd64: /usr/share/gdb/auto-load/usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21-gdb.py libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/share/gdb/auto-load/usr/lib/i386-linux-gnu/libstdc++.so.6.0.21-gdb.py libc6:amd64: /lib/x86_64-linux-gnu/libm.so.6 libc6:i386: /lib/i386-linux-gnu/libm.so.6 libgcc1:amd64: /lib/x86_64-linux-gnu/libgcc_s.so.1 libgcc1:i386: /lib/i386-linux-gnu/libgcc_s.so.1 libc6:i386: /lib/i386-linux-gnu/libpthread.so.0 libc6:amd64: /lib/x86_64-linux-gnu/libpthread.so.0 libc6:i386: /lib/i386-linux-gnu/libc.so.6 libc6:amd64: /lib/x86_64-linux-gnu/libc.so.6 ron@ron-desktop ~ $