For any $a \in \Bbb{Z}$, can we always find two prime numbers $p, q$, such that $p - q \in (a)$?
primes whose difference is a multiple of $n$
jQuery appended table adds closing tag at the end of the text automatically. Why? Motivating question:    I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \ldots $$  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions" $\subset$ $\ldots$  and that at each step you're given a multiplication table that tell how the elements interact. As you move up the ladder, certain "nice" properties are lost: ordering,  commutativity, associativity, multiplicative normedness, etc... Given the multiplication table, you can show that these properties don't hold.   Eric Naslund noted that "the first 4 are very special as they are the unique 4 normed division algebras over ℝ", no surprise then that these $2^n$-ions have found quite a bit of use. I'm interested in the sequence itself however, irrespective of how useful a $2^{256}$-ion might be (ducenti-quinquaginta-sex-ion?).   I feel like something deeper is going on here though that I don't understand. Why are these particular properties lost at each step? Is it possible to quantify the process such that, at the $2^n$-ion you can say something about the symmetry of the multiplication table*?  * I'm making an ansatz that there is a connection between the symmetry of the multiplication table and these "nice" properties. Why do backgrounded processes sometimes stop spontaneously? Sometimes, some time after I've backgrounded a process with bg in bash, when I press Enter in the same shell to redisplay the prompt (just to check that I'm still in bash when some output from the background process has been displayed), the background process seems to stop spontaneously.  If I do bg again the same problem recurs.  The only way to fix it seems to be fg.  Why does this happen? Getting coordinates by clicking on QGIS Canvas with PyQGIS? Are there any procedures when working with QgsMapCanvas (pyqgis, qgis) in order to obtain a pair of coordinates by clicking on the map canvas?   Are there any sample Python codes that demonstrate this? 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! Why doesn't grep using pipe work here? I have the following command:  find / -name libGL.so.1   Which returns lots of lines with "Permission denied". I want to exclude such lines, so I added the following:  find / -name libGL.so.1 | grep -v 'denied'   But the output is the same - my grep -v 'denied' is not filtering out the lines with Permission denied. I've tried many variations, looked over grep tutorials, but I cannot figure out the problem. Any suggestions? “Fire” a weapon before firearms existed? Did the verb “fire a weapon” exist before the actual introduction of firearms on battlefields?  More specifically, does it make sense for a creative work to have archers (or whatever ranged weaponry) be told to “fire!”, when the world they live in has not yet seen firearms?  It seems some kind of an anachronism to me, since before firearms, “fire” would never propel any projectile...  I've seen several movies do it; I can't remember them all, but for instance I verified it in The Lord of the Rings trilogy (left), Kingdom of Heaven (top right), and more recently Frozen (bottom right):    The first two do it several times (with Kingdom of Heaven also using it on ballistae, and The Return of the King even having Aragorn ask Legolas to ), whereas firearms never appear on the battlefields they feature (Kingdom of Heaven is set in the 12th century; as for The Lord of the Rings, apart from one occurrence of a witchcraft-ish bomb, it's archery and medieval siege engines all the way).  Frozen may be debatable, happening probably somewhere in the 18th or 19th century, but since we only see swords, spears and the like, it made me flinch to hear crossbowmen use “fire!”.  I've also noticed other movies avoiding this, using “loose!” instead, such as Troy (left) and Gladiator (right):    This makes me think it would be the right thing to say instead... although I'm still wondering why not use “shoot”, which seems simpler to me. Why are there so many different ways to measure disk usage? Reviewed suggested edit view for title-only edits takes body of last revision? Yesterday I was pointed at some rejected suggested edits. Some of the corrections were valid, so I edited the posts accordingly, while making several further edits. Looking at the suggested edits now however, it seems that for those posts where only changes to the title were originally suggested, the post mortem suggested edit view now incorporates the body of the latest revision. (Or at least it appears like that to me)  Take for example . If you take a look at  of that post, you will see my later edits. The edited body seems to have been incorporated in the suggested edit view. For edits where originally changes to the body were suggested, there is no such confusion.   Why does this matter? In the case of some post mortem discussions or explanations it might help to see the actual situation as it was when the edit was suggested. Especially in the case of edits that might be considered too minor.   So have I stumbled upon a bug here? Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$ Someone had posted a question on this site as to what would be sum of digits of $999999999999^3$ (twelve $9s$ ) equal to?   I did some computation and found the pattern that sum of digits of  $9^3 = 18$,  $99^3 = 36$,  $999^3 = 54$ and so on. So, I had replied that sum of digits of  $999999999999^3 = 12 \cdot 18 = 216$.  Can anybody help me prove this, that sum of digits of  $(\underbrace{999\dots9}_{n\text{ times}})^3 = 18n.$ Let $\mathbf{Y}$ be a random vector. Are $k$th moments of $\mathbf{Y}$ considered? I am self-learning on linear model theory right now, and one thing I find surprising is that although $\mathbb{E}[\mathbf{Y}]$ is defined for a random vector $\mathbf{Y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}$, there is no mention of further moments besides the covariance matrix.  Google searching hasn't turned up much. Are $k$th (raw) moments of $\mathbf{Y}$ considered, or is there a different idea I don't know about?  I am learning from the text  (the TOC starts in p. 17 of the linked file). By "considered," what I mean is is there such a thing as $\mathbb{E}\left[\mathbf{Y}^k\right]$, and if so, how would such a concept be defined? The book I have only covers the first raw moment, and I find it a bit strange that there is no mention of how to define $\mathbb{E}\left[\mathbf{Y}^k\right]$ given my experience in univariate probability, nor do I have the expertise to define it.  Furthermore, if $\mathbb{E}\left[\mathbf{Y}^k\right]$ isn't defined, is there perhaps a related concept that I don't know about that is used instead? I have an excel spreadsheet with header and survey drill data. Header data contains hole ID, and location coordinates, and the survey data contains related downhole survey with Distance, Azimuth and Dip values.   Since I know the hole location and surface elevation, I would like to be able to convert the survey table to XYZ coordinates as well.  Does anyone has a function / procedure / example?  (VB &amp; ArcObjects)  Header Data:    Survey Data: Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero? Embed .mov file via 'Add Media' not working A recent downvote drew my attention to my  four years ago to . I edited the answer to update the links and to (very slightly) change the tone of the answer. I was in a very bad mood at the time of the answer.  Still, reading the comments, I think I was largely correct: it's better for the site if we encourage people to do a little research before posting. Maybe even just the "two minutes of searching" that I refer to in the answer.  I was considering deleting the answer, but decided to bring the old debate back to Meta, instead.  "Was I right, or was I right?" What's the simplest way to list conflicted files in Git? The only solutions that I know of till now are $(x,y) = (1,1) \space , (19,7)$. We can note that: $$x^2-x+1 = y^3 \implies (2x-1)^2 = 4y^3-3$$ Thus, if odd prime $p \mid y$, then $(2x-1)^2 \equiv -3 \pmod{p}$ and thus, $-3$ is a quadratic residue. This implies that $p \equiv 1 \pmod{6}$ .  How can we further proceed into this problem?  Note: As mentioned in one of the links in the comments, if we instead replace $x$ by $x+1$, we get $x^2+x+1 = y^3$. Thus, the solutions for this is $(x,y) = (0,1) \space, (18,7)$. I've been thinking of animating some dice and while most are easy to make, I'm having some trouble figuring out a way to generate a regular dodecahedron. I could hand build it but it would be potentially uneven. Would using the base of a 5 sided cylinder and then some variation on spin be the best option or is there a better way? Question:  Is it possible to get multiple correct results when evaluating an indefinite integral?  If I use two different techniques to evaluate an integral, and I get two different answers, have I necessarily done something wrong?    Often, an indefinite integral can be evaluated using different techniques.  For example, an integrand might be simplified via partial fractions or other algebraic techniques before integration, or it might be amenable to a clever substitution.  These techniques give different results.  For example, looking over a few other questions on MSE:      From : evaluate $$ \int x(x^2+2)^4\,\mathrm{d}x. $$    Via the substitution $u = x^2+2$, this becomes $$ \int x(x^2+2)^4\,\mathrm{d}x       = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + \frac{32}{5} + C.     $$ However, multiplying out the polynomial and integrating using the power rule gives $$ \int x(x^2+2)^4\,\mathrm{d}x       = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + C    $$     From :  evaluate   $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x. $$    Simplifying the integrand using partial fractions then integrating gives $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x       = \frac{-2}{(x+1)} - \ln|x+1| + C.    $$ Via integration by parts, we get $$ \int \frac{1-x}{(x+1)^2} \,\mathrm{d}x       = \frac{x-1}{(x+1)} - \ln|x+1| + C.    $$     From : evaluate   $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x. $$    Using the substitution $u = \sec(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x       = \frac {\sec^2(\pi x)}{4\pi} + C.    $$ Using the substitution $u = \tan(\pi x)$, this becomes $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x       = \frac {\tan^2(\pi x)}{4\pi} + C.    $$ Change multiple filenames by replacing a character I have multiple files named like that : screenshot 13:25.png Windows struggle to open these files probably because of the ":".  How can I replace it?