There's not much more to the question: If $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial, is there a simple proof that there must be some $x$ for which $f(x)$ is a prime number (positive or negative, e.g. $f(x)=-13$ is OK here)?  By irreducible I mean to imply that $f(x)$ is not $0$ or a unit of $\mathbb{Z}[x],$ as some texts demand of an irreducible element. But to simplify things, just assume $f$ has degree at least $1$ [the degree 0 irreducibles being integer primes anyway, which do represent at least one prime].  UPDATE: Note that an example of Hurkyl's gives an irreducible $f$ not representing any primes. $f(x)=x^2+x+4.$ This was pointed out by Lucien in a comment, and that comment has a link to the question with Hurkyl's polynomial in an answer.
Suppose $f(x)$ is an irreducible polynomial over $\mathbb Z$ of degree $n$. Is it always the case that there exist distinct $x_1,\ldots,x_{2n+1}\in \mathbb Z$ such that $f(x_1),\ldots,f(x_{2n+1})$ are all prime?  So far I've tested a few irreducible polynomials using mathematica, and this has held. The motivation is to use this as a test for irreducibility, as if $f(x)$ is not irreducible then it takes on at most $2n$ prime values, as either factor takes on $1$ or $-1$ at most $2n$ times. (I'd also be interested in learning if this bound can be improved).  This could potentially be expended to other rings of integers, i.e. $\mathbb Z[i]$, but in that case the minimal number of primes needed to certify that $f(x)$ is irreducible would increase as more units appear.
How to restore default group/user ownership of all files under /var? I accidentally changed /var owner/group to my username and then changed it back to root, but not all the /var folders' owners are root, so is there anyway to change back owner/group of files/folders to default state?  Or at least those files/folders that are created by packages? Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial. My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following:   Use a line integral to show that $ |f(z)| \leq A + |z| $ where A = $\max\{1, |f(0)|\}$  Thanks for the help. Computing limit of $(1+1/n)^{n^2}$ How do I set Windows to boot as the default in the boot loader? Google Spreadsheets: IF not working I have problems with my IF command in Google Sheets. If gives #ERROR! all the time. See this simple test:    Not even if I put in the number directly and avoid references:    Am I using the IF wrong? Or what could such a problem be? This example is from a new document. How to install all possible media codecs from console?  Is there a way to load and save them for some cases without Internet connection?  Thanks in Advance! How to align a  to the middle (horizontally/width) of the page How to read contacts on Android 2.0 Proof involving absolute value and maximums Given the definition for any real numbers a and b, the max function is   $\max\{a, b\} =  \begin{cases}                                    a \text{ if } a \geq b \\                                    b \text{ if } a &lt; b \\   \end{cases} $  Lemma: for any two real numbers $a\text{ and }b, a \leq \max\{a,b\}$ and $ b \leq \max\{a,b\}$.  Prove that for real numbers, $a, b$, and $x$, if a $\leq x \leq$ b then $|x| \leq \max\{|a|,|b|\}$.  So far, I have  $x \geq 0$ or $x &lt; 0$ by the law of trichotomy  Case 1: $x\geq 0$  $|x| = x$ by the definition of absolute value  and I'm not sure where to go from there. Any hints? Find the Galois group of $x^3-5$ over $\mathbb{Q}$. In this case, the roots of $x^3-5$ are $\{\sqrt[3]{5},\omega\sqrt[3]{5},\omega^2\sqrt[3]{5}\}.$ I think $\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5})$ is the splitting field of $x^3-5.$ Then, $[\mathbb{Q}(\sqrt[3]{5},\omega\sqrt[3]{5}):\mathbb{Q}]=3,$ I think the automorphisms are: $$id$$ $$\sigma:\sigma(\sqrt[3]{5})=-\sqrt[3]{5}$$ $$\tau:\tau(\omega\sqrt[3]{5})=-\omega\sqrt[3]{5}$$ I'm not sure about what I have done. But how to determine the Galois group based on this? And how to determine the solvability of this polynomial? Is there a way to copy maintenance plans from one SQL Server 2012 to another? Both servers are not accessible via the same copy of Management Studio at the same time (on different networks). Consider the atomic spectrum (absorption) of hydrogen.     The Bohr's model postulates that there are only certain fixed orbits allowed in the atom. An atom will only be excited to a higher orbit, if it is supplied with light that precisely matches the difference in energies between the two orbits.   But how precise does 'precisely' mean. Of course, if we need energy $E$ to excite the electron to a higher energy level, and I supply a photon with just $E/2$ I would expect nothing to happen (since the electron cannot occupy an orbit between the allowed ones). But what if I supplied a photon with energy $0.99E$, or $1.0001E$ or some such number. What will happen then?  I think that the electron should still undergo excitation precisely because the lines we observe in the line spectrum have some thickness. Which means that for a given transition, the atom absorbs frequencies in a certain range.   Is my reasoning correct? If not, why? How does Bohr's model explain this? How about modern theory?   If I'm right, what is the range of values that an atom can 'accept' for a given transition? Why do we need sigma-algebras to define probability spaces? How to correct for small p-value due to very large sample size I am running into a problem where an independent variable, which should have no predictive power on the dependent variable based on domain knowledge, comes out with very small p-value because the sample size is very large(~100,000). If I only use &lt; 5000 data points, then the p-value becomes large enough to support the prior that the variable is insignificant. However, I don't think tweaking the sample size to get to the desired conclusion is a good practice. Is there any procedure to adjust for small p-value simply due to huge sample size? How do I install pillow in Ubuntu 16.04 LTS? I went to  for installing Pillow since the original PIL is a discontinued project and so now not so good to use. But, in the table support for Ubuntu 16.04 LTS is unmentioned. It appears neither in , or in .     Further, just installing pillow is said to be not enough. There are some dependencies that should be installed first which is also unmentioned for Ubuntu 16.04 LTS. Would the installation steps for Pillow for python 3 given for 14.04 LTS be OK for Ubuntu 16.04 LTS?  If not, How do I install pillow in 16.04 LTS for Python 3.6? (There is support for Python 2.) Creating bash scripts that take arguments I would like to use Zapf's Palatino font for the text in my document, which also includes a lot of maths. And I would like a good font for the maths. This is a snapshot of what I am doing.  \documentclass[10pt,letterpaper]{memoir} \usepackage{amsmath} \usepackage{amssymb} \usepackage{mathpple} \usepackage{upgreek} \usepackage{mathpazo} \usepackage{lipsum}  \begin{document}  \chapter{Demo}  Some regular text. \lipsum[1]  Now for some maths.  \begin{displaymath} P_{N-1 + m} = \frac{C}{N-{1} + m} \binom{N_{2} - N_{1}}{m}         \alpha^{m}\beta^{(N_{2} - N_{1}) -m} \end{displaymath}  The number 27 is the sum of the first 5 odd prime numbers: $27 = 1+3+5+7+11$  \end{document}   To me the text and math fonts seem a good fit but the text does not look like Palatino. Why can't variable names start with numbers? I was working with a new C++ developer a while back when he asked the question: "Why can't variable names start with numbers?"  I couldn't come up with an answer except that some numbers can have text in them (123456L, 123456U) and that wouldn't be possible if the compilers were thinking everything with some amount of alpha characters was a variable name.  Was that the right answer?  Are there any more reasons?  string 2BeOrNot2Be = "that is the question"; // Why won't this compile? Is my mobile AMD card being used? Sum of the series:$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\cdots$ I have a question where i couldn't find any clue. The question is  $$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+\cdots$$  I could get the general term as $t_n=\frac{1\cdot3\cdot5\cdot7\cdots(2n-1)}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdots2n}$  I have also tried it to form the sequence in the telescopic form.But couldn't get. Any hint will be appreciated.