Let $p$ be prime. Prove if $p|(a^p - b^p)$, then $p^2 | (a^p - b^p)$  don't know where to begin, any help appreciated
$a$ and $b$ are natural numbers and $p$ is a prime number. Prove that if $a^{p}-b^{p}$ is divisible by $p$, then it is also divisible by $p^{2}$.  My attempt: Based on Fermat's theorem $(a^{p}-a)$ and $(b^{p}-b)$ are divisible by p, therefore their difference is divisible by $p$. I.e. $(a^{p}-a)-(b^{p}-b)=(a^{p}-b^{p})-(a-b)$ is divisible by $p$. Since the left hand side of this equation is divisible by $p$, the right hand side also should be divisible by $p$. Based on the problem's assumption $(a^{p}-b^{p})$ is divisible by $p$ therefore we conclude $(a-b)$ must be divisible by $p$. Now We can factorize $(a^{p}-b^{p})$ $$(a^{p}-b^{p})=(a-b)(a^{p-1}+a^{p-2}b+....+b^{p-1})$$ we have to prove $(a^{p}-b^{p})$ is multiple of $p^2$ which means we have to show the right hand side should be multiple of $p^{2}$. But $(a-b)$ is multiple of p therefore we have to show  $(a^{p-1}+a^{p-2}b+\cdots+b^{p-1})$ is multiple of $p$ and I am stuck here. Any help would be appreciated.
I have inserted an image in my LaTeX document and the image always displays with the file path of the image directly above it.  How can I prevent this file path from displaying?  I am inserting the image like this:  \begin{figure} \centering     \includegraphics{C:/Users/Name/Pictures/image1.png} \caption{a caption} \label{fig:reference} \end{figure}   Here is a working example of document that produces this behaviour:  \documentclass{article}  \usepackage{graphicx} \begin{document} \begin{figure} \includegraphics{C:/Users/Name/Pictures/image 1.png}  \end{figure}  \end{document}   The image was inserted using the TeXnicCenter Insert>Picture function.  When I generate the pdf document, directly above any image (I have inserted multiple images using this technique) the file path displays above the image, in the case above it would show: "file\path\of\image", how can I prevent this from displaying? I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and I see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$.    But the first statement is not normally taken as a definition; instead the second is.    What is the intuition, or even derivation behind defining $A$ and $B$ as independent iff $P(A, B) = P(A)(B)$?    The kind of explanation I am looking for would be one similar to that given by Jaynes for the definition of conditional probability in the first chapter of Probability: The Logic of Science, or even a Kolmogorov axiomatic explanation would help. If temperature makes particles vibrate faster, and movement is limited by the speed of light, then I would assume that temperature must be limited as well. Why is there no limit? What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question! get submit button id A result of Erdős on increasing multiplicative functions Erdős proved that if $f(n)$ is a monotone increasing function from the natural numbers to the positive reals, and $f(n)$ is completely multiplicative, then there exists some constant $C$ such that $f(n)=n^C$ for all $n$.  I have not been able to find a nice proof of this online and was hoping someone could provide a proof. Why can't enum's constructor access static fields and methods? This is perfectly valid with a class, but is not allowed with an enum.  What I'm trying to do is store my enum instances in a static Map. Consider this example code which allows lookup by abbreivation:  public enum Day {     Sunday("Sun"), Monday("Mon"), Tuesday("Tue"), Wednesday("Wed"), Thursday("Thu"), Friday("Fri"), Saturday("Sat");      private final String abbreviation;      private static final Map&lt;String, Day&gt; ABBREV_MAP = new HashMap&lt;String, Day&gt;();      private Day(String abbreviation) {         this.abbreviation = abbreviation;         ABBREV_MAP.put(abbreviation, this);  // Not valid     }      public String getAbbreviation() {         return abbreviation;     }      public static Day getByAbbreviation(String abbreviation) {         return ABBREV_MAP.get(abbreviation);     } }   This will not work as enum doesn't allow static references in its constructor. It however works just find if implemented as a class:  public static final Day SUNDAY = new Day("Sunday", "Sun"); private Day(String name, String abbreviation) {     this.name = name;     this.abbreviation = abbreviation;     ABBREV_MAP.put(abbreviation, this);  // Valid } AND gate made out of transistors letting current pass despite only one input high I recently built a gate with transistors, however my meter reads 1.6 volts when only input B is high, and 2.3 while both are high. How can I fix this? (I used 3 volts instead of 6 and 2k instead of 10k) How can I efficiently drive an LED? I understand that I can not connect an LED directly to a battery because it will draw too much current. Thus, there must be something else in the circuit to limit the current. What options are there? Are some methods more efficient than others? What is a good non-technical introduction to theories of everything? The velocity of the flowing coming out of the balloons? Last day , when i was working on two interconnected balloons , a question was kicking my brains !!! This is the explanation of the question:  First , suppose a system that composed of two spherical membranes filled with air (two balloons have different initial volumes {means that the pressure inside which balloons are different} and the air pressure is 1 a.t.m) . We connect them with hollow tube and a valve. When we open the valve , one of them shrinks and the other one expands (It depends on their pressure) .  So how can we get the velocity of the flowing air between two balloons? (Consider everything but if you have reasons for not considering one of them -for example the ratio of friction in the tube- Don't consider them and just tell me the reason)  Thanks Integrating Knowledge into an existing website New tab in chromium is empty Continuous and bounded variation does not imply absolutely continuous Are The Lazarus Effect and Dune in the same universe? There are similar aspects between Dune, and the technology in The Lazarus Effect.   Noticeable similarities:   Lasguns - this weapon exists in both of the books, and has similar architecture in both books.   Merman and Ixians - are both technologically sophisticated in similar ways. Rely on mechanical technology as opposed to organics or similar.   Is this merely Herbert sticking to his original style? "Do you know where's Linda?" vs "Do you know where Linda is?" How do I make Subversion (SVN) send email on checkins? I've always found checkin (commit) mails to be very useful for keeping track of what work other people are doing in the codebase / repository. How do I set up SVN to email a distribution list on each commit?  I'm running clients on Windows and the Apache Subversion server on Linux. The answers below for various platforms will likely be useful to other people though. I have an org with hundreds of apex classes, all of which have api versions ranging between 30-37. Is there an easy way to bring them all to the latest api version at once without resorting to manual updates? How to check if a macro value is empty or will not create text with plain TeX conditionals? I have a macro which holds some text value, e.g. \macro{some text}. If it is empty, I would like for nothing to appear in the document, but it has some content of any kind which would produce text displayed in the document, I would like a message to appear, "Text: some text".   \macro{} would be considered empty. \def\somedata{}, \macro{\somedata} would be considered empty. \macro{0} would be considered not empty.   I have tried making a TeX conditional which can check if the value is not empty, but nothing I tried works, e.g.:  \ifx #1 {} \else     Text: #1 \fi  \ifx#1{} \else     Text: #1 \fi  \ifx #1 \nil \else     Text: #1 \fi  \ifx#1!="" \else     Text: #1 \fi   What is the correct syntax for creating a conditional in plain TeX with checks if a value is not empty? What does a two overlapping circles symbol mean in an electrical schematic/diagram?