Consecutive numbers with prime factorizations with at least one exponent greater than or equal to 1000
Prime factorizations of consecutive numbers, can they both contains powers of greater than or equal to $1000$?
Why do I have to enter my password to log in? Using the Stack Exchange login, why do I have to enter my password again and again to log in? Wasn’t the whole purpose of OpenID to do away with this?  This doesn’t even only happen when visiting a new Stack Exchange site – it also happens for sites where I already have an account but which I don’t visit regularly. Here’s what I mean:    Of course I don’t remember my password (it’s a long random string). And so I have to go to my password manager and manually copy the password from openid.stackexchange.com’s account into the login form. – And incidentally, this is what my password manager looks like now:    (And, by the way, those are the only entries in my keychain so far, but by far not the only SE sites where I have an account.)  … I feel that this defeats the purpose of site-wide login and OpenID. In the olden times when MyOpenID was still working properly, I’d simply enter my OpenID on the login form, and the software would take care of the rest – namely, redirect to the OpenID service, (automatically) verify that I was authorised to use it, and redirect back to the original website, having logged me in. It was literally one button click for me. It’s substantially more work now, and I’d like to understand why this regression is necessary.  (Note: of course once a site’s password has landed in my password manager I (probably?) won’t have to enter it again. But still: wasn’t site-wide login and OpenID supposed to solve this already?) I'm new to d20. I recently acquired a bunch of D&amp;D books and I'm sure they are all either 3.0 or 3.5, but it doesn't say in the book which version it is. How can I tell which version the book is?  For instance, I have "Complete Warrior - A Player's Guide to Combat for All Classes". It's hard back, published in '03. Is it 3.0 or 3.5?  Another example, I have "Sword and Fist - A Guidebook to Fighters and Monks". Perfect Bound, published in '01. Is it 3.0 or 3.5? Is PCA unstable under multicollinearity? I know that in a regression situation, if you have a set of highly correlated variables this is usually "bad" because of the instability in the estimated coefficients (variance goes toward infinity as determinant goes towards zero).  My question is whether this "badness" persists in a PCA situation.  Do the coefficients/loadings/weights/eigenvectors for any particular PC become unstable/arbitrary/non-unique as the covariance matrix becomes singular?  I am particularly interested in the case where only the first principal component is retained, and all others are dismissed as "noise" or "something else" or "unimportant".  I don't think that it does, because you will just be left with a few principal components which have zero, or close to zero variance.  Easy to see this isn't the case in the simple extreme case with 2 variables - suppose they are perfectly correlated.  Then the first PC will be the exact linear relationship, and the second PC will be perpindicular to the first PC, with all PC values equal to zero for all observations (i.e. zero variance).  Wondering if its more general. Size of files in Windows OS. (It's KB or kB?) One "kilobyte" (KB) is 1024 bytes in JEDEC-standard, whereas the definition has shifted, in most contexts, to mean 1000 bytes (kB) in accordance with SI. To resolve this difference, binary prefixes (kiB) are used.  So we have 3 choices for using prefixes - JEDEC, IEC (both in Binary), and Metric (in Decimal).  My questions are-   What prefix standard does Windows use in showing the file size? (surely it's not IEC standard) Why does Windows OS show sizes of files in KB (using a capital alphabet "K") when it's a small alphabet "k" for a Kilo in SI units?   A capital "K" represents Kelvin in SI system of units.  Am I missing something here in understanding? I built a ZI model and it is producing predicted values that are from a very small range when compared to the observed values. Plus it does not produce any zeros. See the fitted vs. observed graph below.     My model structure is given below. My response is a count variable and the predictor is continuous water level measurements (scaled).   m.u.meanwl.gu &lt;- zeroinfl(GU ~ MeanWLScaled | MeanWLScaled, data = uluabat_wl_fg, dist = "negbin", link = "logit")   I need help to figure out what's wrong and fix it. Thanks in advance. Security Review - CRUD/FLS Checks for every SOQL Query I know this sounds like a standard question, so please bear with me.  Recently our Security Reviews fail due to CRUD/FLS violations. The highlighted codelines include almost every SOQL Query in the application.  Our approach is usually as follows:   In case of displaying data on VF pages we either use sObjects fields or use the platform methods to check the accessibility of the fields In case of domain logic triggered by the user (e.g. by clicking a button on a VF page) we have a helper class that checks every field to be updated before doing the DML operation.   Up until recently this approach has worked fine (and passed pervious Security Reviews). As mentioned above the latest Security Review results seem to require us to check every field in a SOQL query for accessibility even if it is never displayed back to the user.    Let's suppose the following program flow as an example:   Query an object with some fields Perform some modification on said fields Perform an update on the sObject to save the modification using a helper class which checks for the required CUD/FLS rights for the modified fields   As I see it there is no need for checking field accessibility in step 1 in that case (since nothing is displayed back to the user and the dml operation is already protected). Is that correct or am I missing something? So is it really required to do an accessibility check for every SOQL Query in the code?  Another question would be how to handle background operations in that case like data purging. These have either to be running in system mode (which causes issues during the Security Review) or require the job to be started by a user with the appropriate access rights (which can be error prone and is not that user friendly). Is there any best practice regarding that.  Unfortunately I could not find a question targeting exactly that issue. Most of the questions I have found are related to how to perform that check (like  or ). So please apologize in case the question was already answered (I would really appreciate a link to the answer then).    Thanks for your time and help.  Mathias It appears that in PHP objects are passed by reference. Even assignment operators do not appear to be creating a copy of the Object.  Here's a simple, contrived proof:  &lt;?php  class A {     public $b; }   function set_b($obj) { $obj-&gt;b = "after"; }  $a = new A(); $a-&gt;b = "before"; $c = $a; //i would especially expect this to create a copy.  set_b($a);  print $a-&gt;b; //i would expect this to show 'before' print $c-&gt;b; //i would ESPECIALLY expect this to show 'before'  ?&gt;   In both print cases I am getting 'after'  So, how do I pass $a to set_b() by value, not by reference? Does a four-divergence extra term in a Lagrangian density matter to the field equations? Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:     ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\mu-\frac{1}{2}\partial_\mu A^\mu\partial_\nu A^\nu $   $\space\space\space\space=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$        The last term is a four-divergence which has no influence on the field equations. Thus the dynamics of the electromagnetic field (in the Lorentz gauge) can be described by the simple Lagrangian        ${\mathcal L}^{\prime\prime}=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu$     Yes, if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field, indeed according to the variational principle this four-divergence will not influence the field equation.  And actually I calculated the time-derivative dependence of 0-component of $[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$, in which only $[A_0(\partial^0 A^0)-(\partial_0 A^0) A^0]$ could possibly contain time-derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case.  But how can he seem to claim that it holds for a general four-divergence term,The last term is a four-divergence which has no influence on the field equations?  EDIT: I only assumed the boundary condition to be $A^\mu=0$ at spatial infinity, not at time infinity. And the variation of the action $S = \int_{t_1}^{t_2}L \, dt$ is due to the variation of fields which vanish at time, $\delta A^\mu(\mathbf x,t_1)=\delta A^\mu(\mathbf x,t_2)=0$, not having the knowledge of $\delta \dot A^\mu(\mathbf x,t_1)$ and $\delta \dot A^\mu(\mathbf x,t_2)$, which don't vanish generally, so the four-divergence term will in general contribute to the action, $$\delta S_j=\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \partial_\mu j(A(x),\nabla A(x),\dot A(x))^\mu =\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \dot j^0 =\int d^3\mathbf x [\delta j(\mathbf x, t_2)^0-\delta j(\mathbf x, t_1)^0]$$ which does not vanish in general! $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. Question is to prove that :     $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.   I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left multiplication action.  Hint given was to prove that stabilizer of any point contains $\{\pm 1\}$.  To prove Cayley's theorem, stating any group is isomorphic to a subgroup of $S_n$ we take action of given group on a set $A$ having same cardinality.  with that motivation I want to check if there is a Isomorphism then there is a map from $G\times A \rightarrow A$. i.e., $G$ gives a permutation group $S_A$.  I tried in same manner. Suppose $Q_8$ is isomorphic to subgroup of $S_n$ with $n\leq 7$  then it should come from a group action of $Q_8$ on a set of cardinality atmost 7.   Suppose $Q_8$ acts on a set $A$ with possible cardinality at most 7.  $stab(a)=\{g\in Q_8 : g.a=g \forall a\in A \}$  $cl(a)=\{g.a : g\in Q_8\}$  I know number of elements in class of $a$ equals to index of stabilizer.  As $cl(a)=\{g.a : g\in Q_8\}\in A$ i.e., $cl(a)\subseteq A$ and as $|A|\leq 7$ i see that $|cl(a)|\leq 7$.  But, $|cl(a)|=|Q_8:stab(a)|$ for any element $a\in A$.  So, $|Q_8:stab(a)|=|cl(a)|\leq 7$ for all $a\in A$.   So, $stab(a)$ should be non trivial subgroup of $Q_8$ if not then $|Q_8:stab(a)|=8$  non trivial subgroup (proper) of $Q_8$ contains $\{\pm1\}$.  So, In the worst case, $\{\pm 1\}\subseteq stab(a)$ for all $a\in A$.  As $Ker(\eta)=\cap_{a\in A}stab(a)$ (where $\eta$ is the action of $Q_8$ on $A$)   we see that $\{\pm 1\}\subseteq Ket(\eta)$ which means that $Ker(\eta)$ is non trivial.  thus there is no isomorphism (coming from $\eta$) between $Q_8$ and any subgroup of $S_7$.  I would be thankful if someone can check whether my approach is correct or if there is any other simple possible way.  P.S : Usually what i do to see whether two groups are isomorphic or not is to check for cardinality, abelian property, no of elements with same order and so on. But I was having no idea when i fail in all these ways. With this Group actions i could see possibility for getting a precise conclusion on Isomorphisms.I would like to Thank Mr. Jyrki Lahtonen (a user of Math.SE) who made me to get used to Group actions.  P.S $2$: If any thing is wrong in my idea, it is entirely my fault, and if anything is correct in this whole credit should go to Mr. Jyrki Lahtonen I remember reading it while in middle school in English in the United States, so the latest it would have come out was 1998. I don't remember if it was all one author, or even the same universe. These are the stories I remember:   A man finds a way to create a medicine that prevents radiation poisoning (or possibly makes someone fireproof... it's a little confused in my head) and learns that part of the problem is finding a market for it. I vividly remember the inventor telling someone to not "let the doorknob ream you" as it slams shut. A bout of graffiti artists leads to the city buying anti-graffiti paint that dissolves any other paint put on top of it. After learning the futility of trying to paint over it, the protagonist consults with a friend and learns that the anti-graffiti paint will eat away at the paint above it at a steady rate, and will even eat successive layers of the anti-graffiti paint. he uses this property to engineer an animation involving a harlequin thumbing this nose at the officials even as they try to wash it away. Some guys (I want to say a gang) go to a clinic for an illegal nanotech transformation. The viewpoint character gets turned into a woman while a friend gets turned into a dog. After roaming about in their new forms, they return to find the clinic being raided, meaning there's no one to turn them back. I think I remember something bad happening to the one transformed into a dog where the transformation continued, erasing his human thought patterns. There was also a scene where the viewpoint character submitted to being sexually assaulted by his former friends as if it was simply to be expected given his form.   I don't remember if the first or the second story was the actual first story of the collection... I want to say that those are the order that they showed up as in the book. The phrase for when: working in a group to divide responsibility and reduce malicious intent What is average distance from center of square to some point? ART has better performance and better battery life on paper at least.  Should I consider switching to it on my moto g? What are the results I can expect? What precautions should I take? Is it easy to turn back to dalvik? How can I make a div not larger than its contents? Correct user names when tracking /etc/ in git repository and committing as root We use git to track changes in /etc/ on our servers.  Administrators work as root when changing files in /etc/, and thus their commits have author  root &lt;root@machinename&gt;   This is not very satisfying since you cannot see which admin actually did the change.  What can we do to get the real admin names in the git log? I don't think that keeping a local clone of the repository is feasible since we often change interatively until something works, and a change-commit-push-seeError-repeat cycle would not help here. How can I remap Caps lock to both Escape and Control? Looking for a single word for 'blind worship' Is there a single word for 'blind worship' as in worshiping an actor blindly notwithstanding the bad performance? Add and remove update channels in an easy terminal way I know there is a GUI in Ubuntu Software &amp; Updates to enable the update channels   updates proposed backports security   as shown in this screenshot:    I am looking for an easy way to do this from within a terminal using commands such as    sudo apt-add-update enable updates sudo apt-add-update enable proposed sudo apt-add-update enable backports sudo apt-add-update enable security  sudo apt-add-update disable updates sudo apt-add-update disable proposed sudo apt-add-update disable backports sudo apt-add-update disable security   and an additional thing  sudo apt-add-update enable default  sudo apt-add-update disable default     Some examples for a better understanding   An empty sources.list   cat /etc/apt/sources.list  &lt;empty&gt;  sudo apt-add-update enable security  &lt;empty&gt;   One enabled repository (main)   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security main   Two or more enabled repositories in one or two lines   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe   or   deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe   or  deb http://archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security main deb http://archive.ubuntu.com/ubuntu wily universe deb http://archive.ubuntu.com/ubuntu wily-security universe   With deb-src entries   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe deb-src http://archive.ubuntu.com/ubuntu wily main universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb-src http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe deb-src http://archive.ubuntu.com/ubuntu wily-security main universe   With inactive deb-src entries   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe # deb-src http://archive.ubuntu.com/ubuntu wily main universe  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe # deb-src http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu wily-security main universe   The default thing    cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily-security universe  sudo apt-add-update enable default  deb http://archive.ubuntu.com/ubuntu wily universe deb http://archive.ubuntu.com/ubuntu wily-security universe   Only one entry and the disable action   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily-security universe  sudo apt-add-update disable security  &lt;empty&gt;   Different or the same servers for different or the samerepositories, respect each server   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily universe deb http://us.archive.ubuntu.com/ubuntu wily main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily universe deb http://us.archive.ubuntu.com/ubuntu wily main deb http://archive.ubuntu.com/ubuntu wily-security universe deb http://us.archive.ubuntu.com/ubuntu wily-security main   Different Ubuntu releases for different repositories, respect each release   cat /etc/apt/sources.list  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu trusty main  sudo apt-add-update enable security  deb http://archive.ubuntu.com/ubuntu wily main universe deb http://archive.ubuntu.com/ubuntu trusty main deb http://archive.ubuntu.com/ubuntu wily-security main universe deb http://archive.ubuntu.com/ubuntu trusty-security main   PPAs or other package sources (not-Canonical)  in the sources.list?  Ignore! Don't change the protocols, e.g. https, http, tor, ... In general relativity (GR), spacetime is viewed as a differentiable manifold of dimension $D$ with a metric of Lorentzian signature $(-,+,+,...,+)$.  My question is why differentiable? So far, I know if $p$ is a rational prime, then $(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$. $(2)$ If $p\equiv1\mod4$  then $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then $π_1 $ and $π_2$ are primes in $\mathbb{Z}[i]$. $(3)$ $2=(1+i)(1-i)$, then $(1+i)$and$(1-i)$ are primes in $\mathbb{Z}[i]$.  What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$? For example, $-3$ are prime in $\mathbb{Z}[i]$, but not in the above $3$ cases.