Are these the only solutions to $a!b!=c!$? Curious about  on,  $$6!7!=10!\tag1$$ a little Mathematica expt yielded, $$r!(r!-1)!=(r!)!\tag2$$     Q: Are the only non-trivial integer solutions to $\color{blue}{a!b!=c!}$ with $a,b&gt;1$ given by $(1)$ and $(2)$?
Solutions of $p!q! = r!$ The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there?  I came across this about ten years ago; as far as I can tell, it hasn't appeared here before, so I thought that it might be of interest. I'm actually most interested in finding whether there was any progress made since
Rename profile folder on Windows 7/8 Set up a scheduled job? Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$ Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate   $$  \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx.  $$   Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \ni \alpha \ne - 1$.  I've done also the case $n=2$ and, if I'm not wrong, it's $\displaystyle \frac{2}{(\alpha+1)(\alpha+2)}$.  My big problem is that I cannot understand how to deal with the general case. Any ideas?   Thanks in advance. I think I've got it down to the most basic case:  int main(int argc, char ** argv) {   int * arr;    foo(arr);   printf("car[3]=%d\n",arr[3]);   free (arr);   return 1; }  void foo(int * arr) {   arr = (int*) malloc( sizeof(int)*25 );   arr[3] = 69; }   The output is this:  &gt; ./a.out   car[3]=-1869558540  a.out(4100) malloc: *** error for object 0x8fe01037: Non-aligned pointer                          being freed  *** set a breakpoint in malloc_error_break to debug &gt;   If anyone can shed light on where my understanding is failing, it'd be greatly appreciated. jQuery: Javascript throws Error "The operation is insecure" when setting value Calculation of subgroups of $(Z_{12}, +)$ How calculate all subgroups of $(Z_{12}, +)$?  I know that the order of subgroups divide the order of the group, but there is such a smart way to calculate the subgroups of order 6? Transferring Lyx Keyboard Shortcuts I recently updated my version of Lyx from 1.6.7 to 2.0.6. I would like to transfer my old keyboard shortcuts to this new version, but I am having difficulty because I was inconsistent with the way that I set them up:   Some were defined in the specified .bind file,  but others were modified in the lyx preferences menu.   I found that when importing my old .bind file into this updated version of lyx, it imported the keyboard shortcuts that were in the .bind file, but not those that I had added in the preferences menu.  Does anyone know if these keyboard shortcuts that were added via the preferences menu are stored in another .bind file somewhere in the lyx directory? Has anyone else successfully transferred keyboard shortcuts that had been implimented in the preferences menu? Why doesn't hydrogen gas exist in Earth's atmosphere? The root mean square velocity of hydrogen gas at room temperature is: Gas constant: $R=8.31\ \mathrm{J\ K^{-1}\ mol^{-1}}$ Molar mass of hydrogen gas: $M=2.02\times10^{-3}\ \mathrm{kg/mol}$ $$\begin{align} v &amp;= \left(\frac{3\times8.31\ \mathrm{J\ K^{-1}\ mol^{-1}}\times300}{2.02\times10^{-3}\ \mathrm{kg/mol}}\right)^{\frac12}\\ &amp;= 3356.8377\ \mathrm{m/s}\\ &amp;= 3.356\ \mathrm{km/s} \end{align}$$ The escape speed of Earth is $11.2\ \mathrm{km/s}$, which is larger than the root mean square velocity of hydrogen gas. But still, hydrogen gas doesn't exist in Earth's atmosphere. Why? Have I made any mistakes in my calculations? I need to cast single figures (1 to 9) to (01 to 09). I can think of a way but its big and ugly and cumbersome. I'm sure there must be some concise way. Any Suggestions Interpret geometrically the equation The function $f$ is convex when:  $$f(tx+ (1-t)x', ty + (1-t)y') ≤  tf(x,y) + (1-t)f(x', y')$$  for all numbers $0≤t≤1$ and all pairs of points $(x,y)$ and $(x',y')$ in the domain of $f$ (the whole of $R^2$)  How would I interpret this geometrically? Also what is an example of a convex function whose graph is not a plane? Calculating the force required to lift a weight with a screw Windows 8.1 missing font files after restart "They're using a cell phone" vs. "They're using cell phones" The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? Visiting the US after overstaying as a minor 13 years ago? Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$? Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$?  What have I tried:  $$\int_0^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_0^1 \Big(\frac{\ln x}{x - 1}\Big)^2 dx + \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx + \int_1^\infty\Big(\frac{\ln t}{\big(\frac{1}{t} - 1\big)t}\Big)^2dt = 2\int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx$$  From the above statement we can conclude, that the integral does indeed converge, as $0 \leq (\frac{\ln x}{x - 1})^2 \leq \frac{C}{(x - 1)^{\frac{3}{2}}}$, for some constant $C$. However, this new form of the integral is not much helpful in finding its exact value.  I also tried substituting $z = \ln x$ , then $dz = \frac{dx}{x}$. Then the integral becomes reduced to $2\int_{0}^{\infty} \frac{z^2}{e^z(e^z - 1)^2}dz$, which, unfortunately, also does not seem to be any easier. Is there an Amazon.com API to retrieve product reviews? "Stay home" or "stay at home" – which is correct and why? Is it ok to use an alternate account while my main user is restricted? I'm new to Stack Exchange, and my main user only has 16+ rep. And it's been blocked in Code Review. I need to use Code Review today. Should I use an alternate account? Bizarre spacing with 4 $\star$s The spacing between the 3rd and 4th stars is non-existent. Anyone know what's up?    \documentclass[11pt]{article}  \usepackage{amsmath}  \begin{document}  \begin{gather*} \star\star\star\star \\ \star\star\star\,\star \\ \star\star\star\star\star \end{gather*}  \end{document} How to evaluate $\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2} \, \mathrm{d}\theta$