I need to prove that $3^i5^j$ can never be a perfect number. How do I tackle this problem. I know that a perfect number is a positive number which is equal to the sum of its positive divisors excluding the number itself. But how do I prove that the number $3^i5^j$ is not perfect?
Define $\sigma(m) = \sum$ d : d|n.  Prove that $p^j$$q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.  Attempt at Solution: I have shown that $p^k$ can never be a perfect number, and im trying to use the multiplicative property of $\sigma$ to generalize to $p^j$$q^i$
I'm travelling from Madrid to Brisbane, but I have a 9 hours layover at Dubai Airport. I do not plan to leave the airport, and the connection is with the same airline (Emirates). I'm from El Salvador.    Do I need a transit visa? I found this , but I am not sure if this is the same issue.  On , I did some reformatting of the indentation in the code block to make it a bit more readable and it was  for peer review. A few minutes later, it appears the OP made an edit that added some additional code. It looks like my suggested edit was applied when the OP saved his/her edit, and it looks like he got the "credit" for the edit.  I'm not really concerned about the 2 rep points; I'm more curious why this happened and if this is a bug (or limitation) in the editing feature, or if this is known/intended behavior?  Note that this is not a suggested edit that's overridden by a concurrent edit: the recorded edit starts from the post-suggested edit state (and in fact can be identical to it, as in  ). The following is a homework problem:  Prove that between two roots of the function $f(x) = \exp(x) \sin(x) -1$ there must be at least one root of the function $\exp(x)\cos(x) +1$.  I think it is possible to prove it by simply computing the signs of the functions at $\frac{\pi}{2}k, \pi k, \frac{3\pi}{2}k, 2\pi k$. However, I'm looking for a more elegant way. It is intuitively clear that the roots of $f$ are ˋˋconverging'' to the roots of $sin$ and the ones of $g$ to the ones of $cos$. Do you have any hint how to construct a nice argument based on this observation? How to reduce the number of authors above >3 to 'First Author et al., (year)'? Prove that $\int_0^{\infty} \frac{\sin x}{x^p}\, dx$ converges for $0<p<2$ Matching Gmail filters on BCC mail How can I start learning about political philosophy? I'm interested in learning about political philosophy. What are the best ways to start? Are there any websites for beginners? Are there any books such as the "For Dummies" series? Are there any audio or video introductions to familiarize myself with the terminology? Is there a way to find the number of accepted answers you have, other than manually counting each one? I often find  syntax very helpful, e.g. process substitution like in diff &lt;(sort file1) &lt;(sort file2).  Is it possible to use such Bash commands in a Makefile? I'm thinking of something like this:  file-differences:     diff &lt;(sort file1) &lt;(sort file2) &gt; $@   In my GNU Make 3.80 this will give an error since it uses the shell instead of bash to execute the commands. class variable vs instance variable --Python .NET Process.Start default directory? I'm firing off a Java application from inside of a C#  console application. It works fine for the case where the Java application doesn't care what the "default" directory is, but fails for a Java application that only searches the current directory for support files.  Is there a process parameter that can be set to specify the default directory that a process is started in? Math Induction Proof: $(1+\frac1n)^n < n$ So I have to prove:      For each natural number greater than or equal to 3,   $$(1+\frac1n)^n&lt;n$$   My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3&lt;3$$ $$\left(\frac43\right)^3&lt;3$$ $$\left(\frac{64}{27}\right)&lt;3$$ which is true.  Now the inductive step, assume $P(k)=\left(1+\frac1k\right)^k&lt;k$ to be true and prove $P(k+1)=\left(1+\frac1{k+1}\right)^{k+1}&lt;k+1$.   This is where I am stuck because usually you add or multiply by $k+1$ or some similar term. Restoring a backup to an older version of SQL Server When trying to restore a backup to a SQL Server Express 2008 database I got the following error:  Restore failed for Server '...\SQLEXPRESS'.  (Microsoft.SqlServer.SmoExtended) System.Data.SqlClient.SqlError: The database was backed up on a server running version 10.50.1600.  That version is incompatible with this server, which is running version 10.00.2531.  Either restore the database on a server that supports the backup, or use a backup  that is compatible with this server. (Microsoft.SqlServer.Smo)   Is there a way to get a backup which is compatible with the older (in my case 10.00.2531) version from the newer (in my case 10.50.1600) version of SQL Server Express? How do I embed a gather environment within a custom environment? I've tried defining the following custom environment, but I get errors:     \begin{gather} ended by \end{solution}   etc...   Here's a MWE showing my problem:  \documentclass{article}  \usepackage[fleqn]{mathtools}  \newenvironment{solution}{%  &lt;-Solution   \par\addvspace{1em}   \itshape{Solution:}\par\begin{gather}}   {\end{gather}\par\normalfont\hfill\qed\par\addvspace{\medskipamount}}%  \begin{document}   \begin{gather}     testing   \end{gather}    %% This does not work:   \begin{solution}     testing   \end{solution} \end{document} How do you iterate backwards through an STL list? $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$ Prove that $\langle u,v\rangle=0\Longleftrightarrow \|u\|\leq \|u+av\|$.  So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints would be greatly appreciated.  Please no answers, I just want a small nudge to get me in the right direction, I've been stumped on this for a little bit.  Here's what I have so far:  ($\Longleftarrow$)  Assume $\|u\| \leq \|u+av\|$  $\|u+av\|^2=\langle u+av,u+av\rangle=\langle u,u\rangle+ \langle u,av\rangle+\langle av,u\rangle+\langle v,v\rangle=\|u\|^2+\bar{a} \langle u,v\rangle+a\langle v,u\rangle+\|v\|^2=\|u\|^2+ \bar{a}\bar{\langle v,u\rangle}+a\langle v,u\rangle+\|v\|^2$  Thoughts/ideas? What does $(command) & do? Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear? This is an exercise in complex analysis:     Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ such that   $$ \varphi(z_0)=z_0\qquad\text{and   }\qquad \varphi'(z_0)=1 $$   then $\varphi$ is linear.   I'm trying work out the case $z_0=0$ first, in which $$ \varphi(z)=z+\sum_{n=2}^{\infty}a_nz^2. $$ It suffices to show that $a_n=0$ for all $n\geq 2$. If let $$ \varphi(z)=z+a_2z^2+O(z^3) $$ then  $$ \varphi^k(0)=z+ka_2z^2+O(z^3), $$ and $$ \varphi^k(0)=0,\quad (\varphi^k)'(0)=1. $$ If one can show that $\{ka_2\}_{k=1}^{\infty}$ is uniformly bounded, then one at least has $a_2=0$. But I don't know how to go on. Any idea? How can I vertically align elements in a div? Proving Geometric Hahn-Banach Implies Analytic Hahn-Banach I've come across a proof online of the H.B theorem using the goemetric version.   There is a step in the proof which im not sure why is true and it is as follows:   Let $X$ be a linear space and $M \subset X \times \Bbb R$ be a maximal subspace.  Then $M = G(F)$ for $F :X \to \Bbb R$ linear.   Someone can show me why this is true?   Maybe I misunderstood the proof, its the last paragraph of Matrin's answer here -.   Thanks a lot for helping!