The Goldbach Conjecture, its negation and consistency
Could it be that Goldbach conjecture is undecidable?
Base-b Representation of an Integer: Why can I make the assumption about number of terms in expansions in uniqueness part?   Reading the book of Koshy I have come across the below theorem. It states that I can write any integer $N$ in a base $b\ge2$ representation $N=a_kb^k+...+a_1b+a_0$ such that $0\le a_i&lt;b$, $a_k \neq 0$ for some $k\ge0$.  In the uniqueness part, why can I assume the expansions contains the same number of terms ? I must prove that all expansions of $N$ equal the one in the theorem, but if I add terms to the expansion by zero coefficients, well, then the expansion is not in the proper form anymore ? (the greatest term in the expansion is $0$ if I add zero coefficients) .  Can someone help clear this out ?     Why can I assume the expansions contain the same number of terms ? If I add terms to the expansion, well then I've violated the form of the expansion ? Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions. Can I travel to Dublin, Republic of Ireland with UK Tier 2/Schengen visas with Indian Passport Registry key / GPO to disable and block Windows 10 upgrade? Petersen graph is not a Cayley graph How can I show that the Petersen graph is not a Cayley graph?  I don't know very much about Cayley graphs, I know that they are vertex-transitive, but so is the Petersen graph. It probably has to do with the group structure of $\Gamma$ in $Cay(\Gamma,S)$ (which is a group of order $10$, i.e., it is either the cyclic group $C_{10}$ or the dihedral group $D_{10}$). Then I am guessing we go for some sort of contradiction(?). My group theory is a bit rusty, I would very much appreciate someone showing me how to do this one! I want to do research but I'm too old for a PHD I really love research. I started a research project in graduate school for my M.A. in English. My research topic is a very viable and interesting topic for a dissertation. I'll be 60 on my next birthday and I know that many Ph.D programs are traditionally geared toward younger students.I was wondering are there any programs that I can continue this research in without facing the issues of ageism in academia? Show that if terms of a series have ratio less than the ratio of convergent series, then the series converges We have two infinite series $\sum a_n $ and $\sum b_n$, such that $a_n &gt; 0,\ b_n &gt; 0$ and $a_{n + 1}/a_n \leq b_{n + 1}/b_n$ for all $n$. Show that if $\sum b_n$ converges, then $\sum a_n$ also converges.   Any hints on how to prove this? I can sort of picture why this must be true, but I'm not sure how to prove this rigorously. I'm not sure using the ratio test would help here.. After update to OS X 10.11 I started having these bugs:   Finder sidebar text flashing  Very high CPU load:      I have Macbook Pro Retina 13" Mid 2014  I reset NVRAM.  I deleted com.apple.finder.plist, it didn't help.  Also I did this: , no longer flashes. Salutation for two doctors (not married) Are objects in PHP passed by value or reference? Does the std::stack in the C++ STL expose any iterators of the underlying container or should I use that container directly? What is ISCAS85 benchmark? What is ISCAS85 benchmark? I find it in a , is there any simulator for it? Split string every nth character? Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$.   Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of $\frak{g}$ is also a linear transformation. Now i have several questions:   How do i prove that $Aut(\frak{g})$ is actually a lie subgroup of $GL(\frak{g})$? I've tried playing with actions to realize $Aut(\frak g)$ as an orbit or an isotropy subgroup but nothing worked out... Say $H \subset G$ is an abstract subgroup (not necessarily lie). Are the following statements equivalent?   $$\{T_e H \text{  is closed under the lie bracket of } \frak{g} \}$$  $$ \{H \text{is a lie subgroup of } G\}$$   How does the lie algebra of $Aut(\frak g)$ look like? When does $Aut(\frak g)$ $= G$? what does $Aut(\frak g)$ tell me about $G$? I was trying to give the wording "isomorphic groups have the same structure" a precise interpretation, at least for the finite order case. A positive answer to my final question would reach the goal, as it would translate a "$\cong$" between groups into a "$=$" between sets.    For $n$ positive integer, let be:   $I_n:=\{1,\dots,n\}$; $G$, $\overline G$ groups of order $n$; $\psi\colon G \rightarrow \overline G$ isomorphism; $f$, $\bar f$ bijections; $\theta$, $\bar \theta$ embeddings; in general, $\varphi^{(\alpha)}$ the isomorphism between symmetric groups on sets of the same cardinality, defined by $\sigma \mapsto (g \mapsto (\alpha\sigma\alpha^{-1})(g))$, where $\alpha$ is a bijection between the sets; $S_n$ the symmetric group of degree $n$.   Visually:  $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c}    I_n     &amp;   &amp;    I_n      \\     \da{f}         &amp;           &amp;     \da{\bar f}          \\    G &amp; \ras{\psi} &amp;    \overline G  \\     \da{\theta}         &amp;           &amp;     \da{\bar \theta}          \\    \operatorname{Sym}(G) &amp; \ras{\varphi^{(\psi)}} &amp;    \operatorname{Sym}(\overline G)  \\     \da{\varphi^{(f)}}         &amp;           &amp;     \da{\varphi^{(\bar f)}}          \\    S_n &amp;  &amp;    S_n  \\ \end{array} $$     Is it $(\varphi^{(f)}\circ\theta)(G)=(\varphi^{(\bar f)}\circ\bar \theta)(\overline G)$? “He is small enough to walk.” But baby wasn't walking so why didn't they use “small” instead of “small enough”? How to unlock gungeoneers How can I take a photo with everything in focus with my DSLR? I am quite new to DSLRs and one of the first things I noticed was the incredible focus points and appearance of depth that you can achieve. That is great in a lot of scenarios, but not in all. When filming a landscape as a whole, I do not want to have the focus on that single tree, but on the whole skyline.  Is there a way by which you can sort-of "disable" the focus, so that the raw image is recorded without any added blur? I suppose something as simple as this is only a small function you can turn off, but I can't seem to find it in the manual of my camera. Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf shading type '0' For the set of data , the following code can't compile showing the error     Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf   shading type '0'. This may corrupt your pdf!.   \documentclass[border= 5mm]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document}     \begin{tikzpicture}     \begin{axis}[unbounded coords=jump, view={70}{40}]     \addplot3 [surf,shader=interp]      table [x index={0}, y index={1}, col sep=tab] {mySurfaceData.txt};     \end{axis}     \end{tikzpicture} \end{document}    So, what went wrong to cause this?  P.S. the original data without NaN is available , where I would like to skip plotting those points of z=1. What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists?