I have a question:  I proved that If $n \in \mathbb{N}$ is an odd number and $\phi(n)$ is a power of $2$ then $n$ is a product of distinct primes.  Now I need to prove that for every odd $n \in \mathbb{N}$ then  $\phi(n) \neq 2^{32}$.  Can someone give me an advice?
Find all $n \in \mathbb{N}$ such that $\phi(n) = 4294967296=2^{32}$.  I managed to find that $16106127360$, $8589934592$, $10737418240$ and $12884901888$ are solutions for the equation, but I do not know how to find all of them.
Is it possible to train a neural network without backpropagation? Many neural network books and tutorials spend a lot of time on the backpropagation algorithm, which is essentially a tool to compute the gradient.  Let's assume we are building a model with ~10K parameters / weights. Is it possible to run the optimization using some gradient free optimization algorithms?   I think computing the numerical gradient would be too slow, but how about other methods such as Nelder-Mead, Simulated Annealing or a Genetic Algorithm?  All the algorithms would suffer from local minima, why obsessed with gradient? Where can I find other RPG players? Man creates miniature population on different time-scale; uses them to invent the 'absolute protection' dome Short science fiction film about a Midwest farmer who builds a bunker after finding a space plant Blender saves a new file every time When I click save or hit Command + S what appears to be a new version of the document is created. So if I save 2 times, I get 2 documents, filename.blend1 and filename.blend2.  What exactly are these files and why is Blender creating them? Alternative to LibreOffice Base for Ubuntu What Happened to The Latex Symbols In the Libreoffice-Texmaths Equation Editor? I recently purged Libreoffice because it was buggy.  I then reinstalled the latest version from apt and now it's missing the symbols above the equation editor.  Here is the current Texmaths equation editor:   There used to be shortcuts for latex including frac{}{} and various set theory operations, as well as the binomial distribution on the top of the window.  What happened to them and how do I get them back? I just figured that they were in some package and installed all packages useing sudo apt install libreoffice*, but this didn't help after I refreshed the program.  EDIT 0: I'm using Kubuntu 19.04  sarah@LesserArk:~$ sudo apt-cache policy libreoffice-texmaths libreoffice-writer texlive libreoffice-texmaths:   Installed: 0.43-2   Candidate: 0.43-2   Version table:  *** 0.43-2 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status libreoffice-writer:   Installed: 1:6.2.2-0ubuntu2   Candidate: 1:6.2.2-0ubuntu2   Version table:  *** 1:6.2.2-0ubuntu2 500         500 http://us.archive.ubuntu.com/ubuntu disco/main amd64 Packages         100 /var/lib/dpkg/status texlive:   Installed: 2018.20190227-1   Candidate: 2018.20190227-1   Version table:  *** 2018.20190227-1 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status recursion-consecutive numbers what is the number of subsets of the set {k∈N|1≤k≤n} with no two consecutive numbers? The answer says: $$a_n=a_{n-1}+a_{n-2}$$ with the starting conditions: $$a_0=1, a_1=2$$1. why does $a_1=2$? $$$$ 2. How does this recursion was created, I understand that if $a_n$ mean all the places with no consecutive numbers so, $a_{n-1}$ come from the fact that after the first place all those places satisfy this condition, but why there is need to "look" at what happen after the two places? What would you call a person who doesn't want to learn anything new? Where can I ask questions regarding Electronics? Is there a general method (which can be implemented by hand) to finding the $n$-th roots of $2 \times 2$ matrices? Is there a similar method for a general $m \times m$ matrix? (for $n &gt; 1$ and $n\in\mathbb{Z}$) I have a large external with many subfolders that I want to move all videos, pictures, and music to another hard-drive, what would be the best way to do this? To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Simulation software to solve high school level mechanics problems I'm a physics teacher and I would like to find a simulation software that could be used to give exact numerical answers to high school level problems in Newtonian mechanics.  I want the students to first calcultate a numerical answer to a physics problem and then use the simulator to visualize the problem in order to understand it better and verify that the simlator gives the same answer that they got when doing the calculation themselves.  Here are some examples of what kind of problems I want the students to be able to solve with the simulator:   A slope with a specific angle of inclination and a ball starts rolling at a certain point. Determine the speed, angular momentum and total mechanical energy of the ball after a specific time t. A ladder of known mass and length is leaning against the wall at a specific angle. The contact between the ladder and the wall is frictionless. Calcultate the x and y components of the force between the floor and the ladder.   I would like the software to have the following features:  - Simulation in 2D. No 3D simulations are necessary and the software should at least have the option of doing simulations only in 2D.  - Position all objects exactly by writing numbers (just positioning the objects using the mouse is not precise enough).  - Show animations of the time evolution of the system.  - Show forces, velocities and accelerations as arrows.  - Give exact numerical results so that the students can use the software to verify their calculation.  Ideally the software would have a graphical user interface, but if no suitable software with graphical interface exixts, I could also consider a software with text-based inputs. I would consider both free and commercial software.  My question is: Is there any software that would meet these requirements and if there is, where could I get it? Build and strengthen the Stack Exchange community with "crossover questions" between sites The energy stored in a capacitor is $$ U= \dfrac{1}{2} CV^2 $$    So when I have a 1F supercap charged to 1V the energy is 0.5 J. When I connect a second supercap, also 1F in parallel the charge will distribute and the voltage will halve. Then    $$ U = \dfrac{1}{2} 2F (0.5V)^2 = 0.25 J $$  What happened to the other 0.25 J? Before today, I would visit Stack Overflow every day and click somewhere in the upper-left corner (I've forgotten what it said, and it's gone now!) and if I didn't have any notifications I would see a list of interesting questions from various sites in the Stack Exchange network which - I believe - were classified as "hot".  Now this new top bar has appeared, and I have no idea how to get this list any more. I'm not really sure what it was called so my searches have been useless as well. "Baggage" versus "luggage" I have the feeling that luggage is more closely associated with vacation travel, whereas baggage is for general transportation.  Or... are they just exact synonyms? What algorithm is used by computers to calculate logarithms?