Prove there is no rational number whose cube is $13$ I used the idea that if $13$ divides $n^3$ then $13$ divides $n$.   I begin with assuming that there is a rational number x such that $x^3$ = $13$. Since $x$ is rational, we rewrite as $m/n$ where gcd(m,n)=1.  $x^3$ = $13$  $m^3$/ $n^3$ = 13.  Then $m^3$ = 13 $n^3$.   Using the fact that if 13 divides $n^3$, then 13 divides $n$. What would the next step be?
Is the cube root of a prime number rational? The question is: if $P$ is prime, is $P^{1/3}$ rational?   I have been able to prove that if $P$ is prime then the square root of $P$ isn't rational (by contradiction) how would I go about the cube root?
Is there anyway to get tuple operations in Python to work like this:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (4,4,4)   instead of:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (1,2,3,3,2,1)   I know it works like that because the __add__ and __mul__ methods are defined to work like that. So the only way would be to redefine them? Embedding DLLs in a compiled executable Anyone knows what software tool is used to draw these circuit schematics I was reading online and found this site:   I was very curious how this professor drew all these circuit diagrams in his lecture notes (such as this one: ). They just look very very good.   And example is shown below:      To be more specific about what I mean, by "publication quality" I mean the following:   The software must output vector graphics, such as eps, or windows meta files. It would be good to have control over the style of the graphics, such the style of the components, line width, figure size, etc. It would be good to have a graphical user interface. I understand LaTeX is very powerful but it's just too painful to write a 100 line script to generate a circuit diagram.   So in short, I'm not really looking for a schematic capture program. I'm looking for a program that can generate pretty circuit schematic illustrations so that I can put them in papers and presentations.  (I am aware of other posts on this forum, such as this one: .  None of the software mentioned in that post seems to get you publication quality figures.) Why does local fn=$(...) mask the $? status code Two function definitions, the only difference is that the first combines the local storage keyword with the assignment, while the second separates them:  function foo {     local fn=$(mktemp -p /path/does/not/exist 2&gt;/dev/null)     echo $? }  function bar {     local fn     fn=$(mktemp -p /path/does/not/exist 2&gt;/dev/null)     echo $? }  foo bar   This echoes "0" then "1". I expect it to echo "1" then "1". It seems like the value of $? is the result of the assignment to local, rather than the result of the command substitution.  Why does bash 4.2.46(1)-release behave this way? Understand PAM and NSS In the last days I have set up some Linux system with LDAP authentication and everything works fine, but there's still something I can't really understand regarding NSS and PAM, also after a lot of research. Citing:  NSS allows administrators to specify a list of sources where authentication files, host names and other information will be stored and searched for  and  PAM is a set of libraries that provide a configurable authentication platform for applications and the underlying operating system  What I don't understand is how PAM and NSS work and interact together. In  the architecture is explained pretty well: I configure PAM to use pam_ldap for LDAP accounts and pam_unix for local accounts, then I configure nsswitch.conf to fetch information from local files and LDAP. If I have understood correctly LDAP is used twice: first by pam_ldap and then by NSS which is itself called from pam_unix. Is that right? Is LDAP really used twice? But why do I need to configure both NSS and PAM? My explanation is that PAM performs different tasks than NSS and it is used by other programs. But, then, it should be possible to use only NSS or only PAM, as I have read in . So I experimented a bit and I have first tried to delete LDAP from the nsswitch.conf (and the authentication stopped to work as if only pam_ldap is not enough to do the job). Then I re-enabled LDAP in NSS and I deleted it from the PAM configuration (this time everything worked fine, as if pam_ldap is useless and NSS is enough to authenticate a user). Is there anyone who can help me to clarify this? Many thanks in advance. UPDATE I've just tried something now. I removed again all the pam_ldap entries in all pam configuration fields and I have also removed shadow: ldap from nsswitch.conf. As now in all the system there are only the lines: passwd: ldap files and group: ldap files in nsswitch.conf. Well... the login with LDAP users works perfectly, those two lines (plus /etc/ldap.conf) are enough to configure LDAP auth. From my knowledge PAM in independent from NSS, but my tests showed it's not. So I ask myself is it possible to completely disable NSS and use only PAM? Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? Research statement ideas got used by interviewing committee Conjugation: Do we cite authors or papers? When referencing to another work in a scientific paper, do we cite the paper or its author(s)?  This question is intended to clarify the conjugation of the verb that follows the reference -- especially in these cases:   One author, two papers:     Jane Doe (2015a, 2015b) list-s the very specific conditions under which...   -> lists (singular: referencing to Jane Doe) or list (plural: referencing to both papers) Several authors, one paper:     John Doe, et al. (2015) claim-s this and that.   -> claim (plural: referencing to the multiple authors of the paper) or claims (singular: referencing to the single paper). What system monitoring tools are available? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. Reducing voltage with resistors I have been looking around for an easy way to convert 12V to 5V. I have seen some people saying that a simple resistor is all that is needed.  $$ Volts = Ohms \cdot Amps $$ $$ Amps  = \frac{Volts}{Ohms} $$ $$Ohms  = \frac{Volts}{Amps}$$  So applying a resistor will diminish the voltage of the circuit. That should mean that an appropriately sized resistor could simply be placed in the path of a 12V circuit, converting it to 5v.   If this is the case how would one reduce amps? Would series vs parallel make a difference in this area?   I have seen designs that include a regulator IC and some capacitors, but if a simple resistor/fuse/diode setup will do the trick, I would really prefer that. Windows 8 opens notepad on boot- no startup entry found I just upgraded from windows 7 to windows 8 and everything is working fine except notepad keeps opening on boot with a "access denied" window that comes up as well. I've ran MBAM and removed some junk, I've disabled every "file not found" item in autoruns as well as something that had a text/xml title in it and I've ran CCleaner a few times. Nothing is working, any help is much appreciated! Choice of personal pronoun in single-author papers How many files can a windows folder contain?  This question is asked  but then the question text pollutes the question with issues of "rules of thumb"  So to be clear, this question is: How many files can a windows folder contain? Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above? Extract current viewing parameters from a 3D view? I'm Australian and have obtained my Canadian working visa. However rather than flying direct to Canada I am meeting an American friend in LA and driving across the border. I have a return flight booked back to Australia from LA in March of next year too. Will the esta visa will be enough to enter the USA because I have my Canadian working visa and plan to travel there? Or will there be a problem because my return flight is booked later then the 90 days allowed on the visa. Typesetting A when a register is 1, B when 2, etc I have a register \count0=&lt;some-value&gt; and need to get the letter of the alphabet on the index of \count0. That is, if it is 1, I need an A, if 2, a B, etc.  Is this possible?  I'm reluctant sharing an MWE as it seems to overcomplicate things a bit:  \begin{tikzpicture} \node[regular polygon, regular polygon sides=7, minimum size=5cm] (A) {}; \foreach \i in {1,...,7}{     \count0=8     \advance\count0 by -\i     \node[circle, label=above:\scriptsize\the\count0, fill=black, minimum size=4pt, inner sep=0pt] (\the\count0) at (A.corner \i) {}; } \end{tikzpicture}   With the tikz package and the shapes.geometric tikz library.  This draws seven nodes labelled 1 .. 7 on a circle. I need to have A .. G instead of 1 .. 7. Is there an easy way of getting the charset and collation of the DB tables in WordPress without resorting to SQL queries? What is Ambient Occlusion? What is Ambient Occlusion and what are some of it's uses? I see it is used in creating what might be called "shadow catchers" in the following node tree setup: Is it possible to search for down voted questions? I want to search for new downvoted questions to improve or flag them.   I tried in the searchbar: votes:-1 is:question   but votes:XX only works with positive numbers. I can't query for downvoted questions. Is there a work-around to this? If there's not, should this feature be added?