Determine all positive integers $(a,n)$ such that $(a-1)!+1=a^n$
Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
Pacman option to assume "yes" to every question? PrintWriter append method not appending What IDE to use for Python? What IDEs ("GUIs/editors") do others use for Python coding? Lost credit card replaced with new card and new numbers. Credit score affected? I recently left my credit card at a bar so I reported it as lost to my bank, and my bank said I will get a replacement within a few days. For what it's worth, I picked my card up from the bar a few days later and made sure to cut it up.  The new credit card has arrived and has new numbers. Will this negatively affect my credit score?  VISA, Chase Bank, USA. $4 \sin 72^\circ \sin 36^\circ = \sqrt 5$ How do I establish this and similar values of trigonometric functions?  $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$ Why Do Glueballs Have Mass, When Individual Gluons Are Massless? Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists I have to show that set of all $x$ such that $x \in P$ and $x \notin Q$ exists. So, I think I have to assume that sets $P$ and $Q$ already exist. So, with that, I can use  . This axiom is $$ \forall w_1,\cdots, w_n \, \forall A \,\exists B \,\forall x \left( x \in B \Longleftrightarrow [ x \in A \wedge \varphi(x,w_1, \cdots , w_n, A)] \right) $$ So, I will let $\varphi(x, Q, P) = x \in P \text{ and } x \notin Q $. Also, I will let $A = P$ and  $w_1 = Q $. So using universal instantiation, I get the following $$ \exists B \,\forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge \varphi(x, Q, P)] \right) $$ And using, existential instantiation, there exists a set $B$ such that $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge \varphi(x, Q, P)] \right) $$ Using the definition of $\varphi(x, Q, P)$, this can be simplified as following $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge x \in P \wedge x \notin Q] \right) $$ $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge x \notin Q] \right)$$ $$ \forall x \left( x \in B \Longleftrightarrow x \in \{ x \,| x \in P \text{ and } x \notin Q \} \right)$$ Now, using , it will follow that $$ B = \{ x \,| x \in P \text{ and } x \notin Q \}  $$ And, since $B$ exists, this means that $\{ x \,| x \in P \text{ and } x \notin Q \} $ also exists. Is the proof good ? Do I need a transit visa for a layover in the United Arab Emirates? Do I need a transit visa for a layover via airports in the UAE, such as Dubai International Airport or Abu Dhabi International Airport? If so, what are the requirements for such a visa? Can I get the transit visa on arrival? Is there a way to find the number of accepted answers you have, other than manually counting each one? Would depositing large cash amounts cause problems? I saw  recently where a man withdrew $600,000 USD in cash from his bank.  I'm sure many folks here would advise against this move for obvious (and perhaps not-so-obvious) reasons.   My question: suppose you did this and then, some months later, changed your mind and wanted to re-deposit with a bank.  I assume the gent cannot simply walk into a bank with $600,000 USD and deposit it, right?  Wouldn't money-laundering alarms be raised?  Wouldn't the IRS like to know where the money came from? I'm using the beamer miniframes theme (with subsection=false) and the compress option. Currently the navigation shows the dots for each subsection side-by-side. I would prefer if there would be a horizontal space (e.g. with the width of a dot) between the dots of each subsection. Is this possible? Right now it looks like:  Section 1 oooooo  And I would prefer it to look like:  Section 1 ooo ooo  MWE: \documentclass[compress]{beamer}   \useoutertheme[subsection=false]{miniframes}   \begin{document}   \section{Sec 1}   \subsection{}   \begin{frame}[t]     hi   \end{frame} \subsection{}   \begin{frame}[t]     hi   \end{frame} \end{document}` Can you play a Progress Card even when you get nothing from it? If I have a Progress Card for which initial preconditions do not hold, will I still be able to use it in order to discard it back to the pile? The goal is to prevent a spy from stealing it.  A few examples:   I am leading in victory points but I want to use Wedding card. I have Irrigation but I don't have any wheat hexagons. I have Blacksmith but I don't have any ore hexagons. Why can't I sort questions by "votes" or "unanswered" without logging in? On the "Questions" page of each Stack Exchange site, there are links at the top to sort the questions by "newest", "frequent", "votes", "active", and "unanswered".  At least that is what it looks like if I am logged in. If I am on a site that I don't have an account for, or if I am not logged in, the only one I see is "newest".  This seems to have been done recently. I could swear I saw the other links, even when I was not logged in, until a few weeks ago, or maybe later than that.  Why was this change made? Can I visit the UK with an EU permanent residence card? Add layer to a QGIS group using Python The locus of the intersection point of two perpendicular tangents to a given ellipse In what cases should I use a comma before "or"? Functions of Independent Random Variables I know classical physics, quantum mechanics, special relativity, and basic nuclear physics. I would like to get into some particle physics.   Where can I find a good introduction? It might be useful to segment books by whether they require quantum field theory or not. What is the easiest way to strip a desktop edition to a server edition? We have installed Ubuntu desktop edition on our development server. Now that we have it in a data center we would like to strip it down to a server edition.  Is there an easy way of doing so rather than just going in and uninstalling packages by hand?