Problem:Find all primes $p$ such that $p^3-4p+9=x^2$, where $x$ is a positive integer.  My Attempt: I have made the following observations while trying tot solve this problem:    $p=2$ yields $x=3$. For $p&gt;2$ the following is true: $x^2\equiv 0\pmod 4$ and $p\equiv 3\pmod 4.$  It seems that $p=2,7$ and $11$ are the only primme numbers that satisfy the conditions of this problem. Below are the values of $p^3-4p+9$ evaluated for $1\leq p\leq 100.$   9  24  114  324  1296  2154  4854  6792  12084  24282  29676  50514  68766  79344  103644  148674  205152  226746  300504  357636  388734  492732  571464  704622  912294  I would like to know how one can prove that $p=2,7$ and $11$ are the only solutions to the above-mentioned equation.
Find all primes $p$ such that  $ p^3-4p+9 $ is a perfect square.   I tried a few different values for $p$, namely $2,3,5,7,$ and $11$. The prime $p =2,7,11$ all worked but $p =13$ didn't so it makes me wonder. How can I find all primes such that it is a perfect square?  EDIT: it turns out this is problem P25 in post number #63 at
Does visiting Canada from USA under ESTA reset the visa waiver? How do you get the caret position in a &lt;textarea&gt; using JavaScript?  For example: This is| a text  This should return 7.  How would you get it to return the strings surrounding the cursor / selection?  E.g.: 'This is', '', ' a text'.  If the word “is” is highlighted, then it would return 'This ', 'is', ' a text'. How do I run a program with commandline arguments using GDB within a Bash script? iBooks library like effect Multiplicative inverse questions This is my first time posting on Math exchange. I have been self-teaching myself Mathematics and recently started learning some Algebra. I was posed the following questions, in which I would like to answer before I proceed, but having some trouble finding proper solutions. So the format will be question, and my answer.      What name do we give an element $a \in \mathbb{Z}_n$ that has a multiplicative inverse?   A modular multiplicative inverse of an integer $a$ is an integer $x$ such that $ax \cong 1 (\mod m)$. This means that the remainder after dividing $ax$ by integer $m$ is equal to $1$.      Give an example, for some $n \in \mathbb{N}$, of an element of $\mathbb{Z}_n$, that does not have a multiplicative inverse?   This question is throwing me off because - does the $n$ we speak of matter for both $n \in \mathbb{N}$ and $\mathbb{Z}_n$ ot solely $n$? My guess here would be $0$ because we cannot have an inverse of $0$.      Prove that multiplicative inverse are unique; an element $a \in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Suppose that an element $a\in\mathbb{Z}_n$ has an inverse $a'$. Next, suppose that $a^*$ is also an inverse of $a$. Then, $a'$ is a solution to $a \cdot_n x=1$, and similarly $a^*$ has a solution to $a \cdot_n x=1$.   Lemma: Suppose $a'$ is multiplicative inverse of $a$ in $\mathbb{Z}_n$, then for any $b \in \mathbb{Z}_n$ the equation $a \cdot_n x=b$ has a unique solution, such that $x = a' \cdot_n b$.   By Lemma, equation $a \cdot_n x = 1$ has a unique solution, namely $x=a'=a^*$. Therefore, an element $a\in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Thank you for the help. I would love any constructive feedback for these. If you can help me elaborate on some ideas, that would be really helpful also.  Thank you very much! $dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is. LaTeX figures side by side Can we create a subsection of the References Cited section with different reference numbering? I need to add some more publications. I do not want to change the previous reference numbers, so I need to create new citations with difference reference numbering like [p1], [p2], etc. Can LaTeX do that? How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime? I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on, Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. problem with surfaces on tkz-fct i am using this wonderful tool that tkz-fct is for creating nice documents but I have a weird result from recently. This is my document :  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{calc} \usepackage{color} \usepackage{pifont}                     %pour utiliser la police Ding \usepackage[left=1.5cm,right=1.5cm,top=2cm,bottom=2cm]{geometry} %marges du document \usepackage[french]{babel} \usepackage[usenames,dvipsnames]{xcolor} \usepackage[L]{thmbox} \usepackage{eurosym} \usepackage{enumitem}               %pour modifer itemize et enumerate \usepackage{amsmath,amsfonts,amssymb} \usepackage{mathrsfs}               %pour utiliser mathscr \usepackage{array,multirow,makecell} \usepackage{fancyhdr}               %package pour modifier les en-têtes \usepackage{fancybox}               %pour boîtes spéciales \usepackage{tikz,tkz-tab,tkz-fct} \usepackage{tikz-dependency}        %permet de créer des schemas \usepackage{sectsty}                %permet de modifier les styles des sections (voir \sectionfont l.21) \usepackage{fourier-orns} \usepackage{shadethm}               %pour theoremes en gris \setcellgapes{1pt} \makegapedcells \newcolumntype{R}[1]{&gt;{\raggedleft\arraybackslash }b{#1}}   %pour redéfinir les colonnes dans tabular \newcolumntype{L}[1]{&gt;{\raggedright\arraybackslash }b{#1}} \newcolumntype{C}[1]{&gt;{\centering\arraybackslash }b{#1}} \renewcommand{\thesection}{\Roman{section}} %permet de changer la numérotation en chiffres romains des sections \newshadetheorem{defi}{Définition}  %crée un nouveau type de théorème \newshadetheorem{thm}{Théorème} \newshadetheorem{ppte}{Propriété} \sectionfont{\sectionrule{3ex}{3pt}{-1.5ex}{1pt}} %mettre un trait de séparation dans les sections (package sectsty) \pagestyle{fancy} %style en-têtes \lhead{} %en-tête gauche \chead{Lois à densité} %en-tête centrale \rhead{T\up{ale}} %en-tête droite \begin{document} \begin{tikzpicture}      \tkzInit[xmin=-4,xmax=4,ymax=1]      \tkzDrawX[label={},noticks]      %\tkzDrawY      \tkzFct[domain=-2:2]{exp(-x**2)}      \tkzDrawArea[color=blue, domain = -0.5:1]      \tkzText(-1,0.75){$\mathscr{C}_{f}$} \end{tikzpicture} \end{document}   But from now, it doesn't produce the result I would like as the surface start from the point (-0.5,f(-0.5)) and not as an integral from -0.5 to 1. This is a picture of the result :   Which is obviously not what I would like and this extends now to all my "old" docs which were "normal" before. I don't think I have done anything special except updating my Tex distribution ... If someone has an idea ?? highest power of prime $p$ dividing $\binom{m+n}{n}$ Are $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ different? I have seen the claim that $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a cute application of the hairy ball theorem). How would I prove that they are indeed not diffeomorphic (or better, not homeomorphic)? Why is it possible to conclude everything from a false statement? Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) As Joel points out in , in  (aka: K &amp; R), there is mention of this property of arrays in C: a[5] == 5[a]  Joel says that it's because of pointer arithmetic but I still don't understand. Why does a[5] == 5[a]? UK coastline shapefile? java.io.Console support in Eclipse IDE I use the Eclipse IDE to develop, compile, and run my Java projects. Today, I'm trying to use the java.io.Console class to manage output and, more importantly, user input.  The problem is that System.console() returns null when an application is run "through" Eclipse. Eclipse run the program on a background process, rather than a top-level process with the console window we're familiar with.  Is there a way to force Eclipse to run the program as a top level process, or at least create a Console that the JVM will recognize? Otherwise, I'm forced to jar the project up and run on a command-line environment external to Eclipse. An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root? Problem: Is there an irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root for $p$ prime? If there is, what is the minimal degree possible?   I can only prove that $x^2-c$ is impossible, by quadratic reciprocity and Chinese remainder theorem. Even the case of $a x^2 - c$ is unknown to me.  Meanwhile, if $f$ is not required  to be irreducible, but only have no root in $\mathbb{Z}$, then $(x^2-a)(x^2-b)(x^2-ab)$ is a solution for non-sqaure $a,b,ab$, since if both $a,b$ are not squares in $\mathbb{Z}/p\mathbb{Z}$, then $ab$ is a square. Yet I still don't know if this is of minimal degree.  Also, it is natural to pose generalizations   When $p$ is not necessarily prime (equivalently for all prime powers). When $p$ is odd prime. When $p$ represents sufficiently large primes (equivalently all but a finite number left). How do I parse an ISO 8601-formatted date? I've found  many queries to check edit counts, and I get different results for each one. Which one is the most accurate and up-to-date query? Or is there any way to see the edit count in stats?