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This sandbox is intended for saving drafts of long, complex posts, especially posts whose composition takes a long time. It serves to localize to one thread the front-page "bumps" caused by edits to drafts of such posts, so that they may be easily ignored. Also, it helps to guard against losing longly-composed posts due to system crashes.

When you are happy with your draft here, you may simply copy the code and paste it to the desired location.

Proper Use of the Sandbox

  1. Do not post a new answer! We wish all the answers on this page to be owned by the Community user (so that only a non-sentient bot is informed of edits to these answers). Posting a new answer will make you the owner, meaning that you will be notified whenever another user makes an edit to that answer.

    The sandbox has been "wiki locked" to prevent the creation of new answers. There are more than enough existing answers for users to edit over, and this will greatly reduce the frequency at which we request that the answers be disassociated from specific users.

  2. Do not delete answers! Deleting seems like a reasonable option, but there are no "hard deletions" on Stack Exchange, and users with sufficient privileges will still see your supposedly deleted postings. Deleted answers will be undeleted and cleared for the use of others.

  3. Do look for an answer which indicates that it is free and then edit it to your heart's content. If none appears available, take over the one that has been left unchanged the longest (which will appear at the bottom of the page if you order answers by "activity").

  4. Do not expect your draft to remain untouched for days. There are no guarantees that your draft will be the latest revision if you return days later. While users will try not to step over others' toes, it may happen that an unfinished draft is edited out. Your draft will, however, still exist as a revision of the answer it was made in. If your drafting is expected to take place over a longer period of time, either

    • take note of the URL of the answer provided by clicking the share button, or
    • save a copy of your draft locally (or even "in the cloud").

  5. Do clear your draft when you are finished. This includes removing all $\LaTeX$ from your answers. Replacing all code with a simple statement like

    This answer is free for anyone to use

    is sufficient. Periodically users may go through and free up answer slots that have not been edited in, say, over one month. But you can aid in the smooth running of this sandbox by clearing away your drafts when you are finished with them.

  6. Do not "claim" multiple answers concurrently. Since this post is closed, the answers are a limited resource. If you really must compose several long, complex posts at the same time, you can still use a single answer, separating the different drafts using Markup: horizontal rules (---) and/or headings (# Header 1 #) are natural choices.

  7. Do not create new such sandboxes. The point of having a unique such sandbox is that it minimizes the noise on the front page when the sandbox is edited. If there were multiple sandboxes they will frequently occupy numerous front page slots, pushing other topics off the front page, and increasing noise.

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locked by user642796 Jan 12 '17 at 6:11

This question's answers are a collaborative effort: if you see something that can be improved, just edit the answer to improve it! No additional answers can be added here

Read more about locked posts here.

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    $\begingroup$ I have added a [sandbox] tag to allow people ignore it more easily (via software support of ignoring tags), and since it seems that we have two sandboxes now, a tag may seem a bit more in place here. $\endgroup$ – Asaf Karagila♦ Jul 18 '12 at 8:35
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    $\begingroup$ (+1) For thinking outside the (sand)box. $\endgroup$ – cardinal Jul 18 '12 at 19:40
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    $\begingroup$ At the suggestion of the moderators, I have gone and changed the associated owners of all the answers here to the Community user. This way, the original owners will not receive excess pings for each time another user uses the draft space for their work. Enjoy! $\endgroup$ – Grace Note♦ Oct 5 '12 at 14:45
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    $\begingroup$ To prevent crashes I've found the "Bookmarks to disable/enable MathJax", provided in here, pretty useful. $\endgroup$ – leo Dec 17 '12 at 18:03
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    $\begingroup$ PSA: Between the creation of this sandbox (in July 2012) and today (December 2015), technology has advanced. Something like StackEdit (or others, it's simply the only one I know) essentially solves all the limitations of this sandbox. You can have multiple concurrent drafts, you don't have to worry about polluting meta's front page, you can leave your draft untouched for days and expect it to still be there, you don't have to explicitly clear up your draft when you're done... Maybe someday we can get rid of this outdated crutch. $\endgroup$ – Najib Idrissi Dec 2 '15 at 14:07

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\begin{align} 2^{10}10^{2018}+7^4 &= 2^{10}100^{1009}+(50-1)^2 \\ &= 2^{1019}50^{1009} + 50^2 -2\cdot50 + 1 \\ &= 50(2^{1019}50^{1008} + 48)+1 \end{align}

factoring $50(3n+2)+1$ \begin{array}{r} 50( 2)+1 &= & 101 \\ 50( 5)+1 &= & 251 \\ 50( 8)+1 &= & 401 \\ 50(11)+1 &= & 551 &= &19\cdot29 &= &(24-5)(24+5) \\ 50(14)+1 &= & 701 \\ 50(17)+1 &= & 851 &= &23\cdot37 &= &(30-7)(30+7) \\ 50(20)+1 &= &1001 &= &7\cdot 11 \cdot 13 \\ 50(23)+1 &= &1151 \\ 50(26)+1 &= &1301 \\ 50(29)+1 &= &1451 \\ 50(32)+1 &= &1601 \\ 50(35)+1 &= &1751 &= &17\cdot 103 &= &(60-43)(60+43) \\ 50(38)+1 &= &1901 \\ 50(41)+1 &= &2051 &= &7\cdot 293 &= &(150-143)(150+143) \\ 50(44)+1 &= &2201 &= &31\cdot 71 &= &(51-20)(51+20) \\ 50(47)+1 &= &2351 \\ 50(50)+1 &= &2501 &= &41\cdot 61 &= &(51-10)(51+10) \\ \end{array}

We may as well use the points $P_k = (\cos(\theta k), \sin(\theta k))$ where $\theta = \dfrac{2\pi}{5}$ and $k \in \{0,1,2,3,4\}$.

$$\frac{XA+XB}{XC+XD+XE}$$

Let $\theta = \dfrac{2\pi}{5}$. Then we can define \begin{align} A &= (\cos(2\theta),\sin(2\theta)) \\ B &= (\cos(3\theta),\sin(3\theta)) \\ C &= (\cos(4\theta),\sin(4\theta)) \\ D &= (\cos(0\theta),\sin(0\theta)) \\ E &= (\cos(1\theta),\sin(1\theta)) \\ \end{align}

$\begin{array}{ccc} \theta & \cos \theta & \sin \theta \\ 0 \cdot\dfrac{2 \pi}{5} & 1 & 0 \\ 1 \cdot\dfrac{2 \pi}{5} & \dfrac{\sqrt 5 - 1}{4} & \sqrt{\dfrac{5+\sqrt 5}{8}} \\ 2 \cdot\dfrac{2 \pi}{5} & -\dfrac{\sqrt 5 + 1}{4} & \sqrt{\dfrac{5-\sqrt 5}{8}} \\ 3 \cdot\dfrac{2 \pi}{5} & -\dfrac{\sqrt 5 + 1}{4} & -\sqrt{\dfrac{5-\sqrt 5}{8}} \\ 4 \cdot\dfrac{2 \pi}{5} & \dfrac{\sqrt 5 - 1}{4} & -\sqrt{\dfrac{5+\sqrt 5}{8}} \\ \end{array}$

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This entry is available for anyone who knows how to enjoy a bit of Lorem ipsum in life from time to time.

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This slot is as vacant as the expression of someone who's been staring at the same line of a proof for three hours.

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This entry is available for anyone to commandeer since Oct. 17th, 2018.

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This space free for whoever wants it

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This answer is available for anyone to use.

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This slot is open for anyone to use.

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This space is freely open to everyone.

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This answer is available for anyone to use.

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This answer is free for anyone to use.

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This answer is available for use by anyone.

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This entry is available for anyone to commandeer since Oct. 13th, 2018.

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This entry is free for anyone to commandeer since Oct. 27th, 2018

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Function $f$ is defined as $$f(x)=\frac{1}{(x-1)^3(x^3+2x^2+x+1)}\tag 1$$ Note that $$x^3+2x^2+x+1=(x+1)(x^2+x+1)\tag 2$$ and so $$f(x)=\frac{1}{(x-1)^3(x+1)(x^2+x+1)}\tag 3$$ and also $$f(x)=\frac{1}{(x-1)}\cdot\frac{1}{(x^2-1)}\cdot\frac{1}{(x^3-1)}=\sum_{i=0}^\infty x^i \cdot\sum_{k=0}^\infty x^{2k} \cdot\sum_{j=0}^\infty x^{3j}$$

$$\frac{1}{(x-1)^3(x+1)(x^2+x+1}=\frac{A_1}{(x-1)}+\frac{A_2}{(x-1)^1}+\frac{A_3}{(x-1)^3}+\frac{B}{(x+1)}+\frac{Cx+D}{(x^2+x+1)}$$ $$1=A_1(x-1)^2(x+1)(x^2+x+1)+A_2(x-1)(x+1)(x^2+x+1)+A_3(x+1)(x^2+x+1)+B(x-1)^3(x^2+x+1)+(Cx+D)(x-1)^3(x+1)$$

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  • $\begingroup$ Oh god, I still remember having tests that required computing these king of generating functions by hand. If result was correct, the method was verified, otherwise no points. $\endgroup$ – enedil Feb 11 at 21:53
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It's been a long time since I last did this stuff, but nobody better seems to have turned up, so I'll have a go.

I'll give the box $\square \phi$ its usual meaning here: "when I translate $\phi$ into the formal system using Gödel numbers, then there is a collection of mechanical transformations inside the formal system that together form an internal-system-proof of $\phi$'s quotation". I will also refer to the formal system in question in the first person, with the pronoun "I".

Then the formula $\square (\square \phi \to \phi) \to \square \phi$ should be read as:

If there is a number which encodes a proof that "having a number which encodes a proof of $\phi$ means I can actually prove $\phi$", then there is a number which encodes a proof of $\phi$.

Or, a bit more loosely,

If I, internally, can prove that my proof of $\phi$ respects the logic in which I operate (so my proof of $\phi$ means that in my surrounding logic, $\phi$ is actually true), then in fact there is a proof of $\phi$ that I will be satisfied with.

This is more concretely understood through its universal quantification, Löb's theorem (which is true in any system rich enough to support Peano arithmetic):

If I can prove within myself that my own proof-checking process respects the semantics of the logic in which I operate, then the logic realises that actually I can satisfy myself of any statement at all.

Specifically, $\square (\square \phi \to \phi)$ means "I contain a proof that my proofs are semantically valid in the surrounding logic", while $\square \square \phi \to \square \phi$ means "it's true in the surrounding logic that if I want to prove $\phi$, it's enough to prove provability-of-$\phi$".

The two statements were very unlikely to be the same: one is talking about proof within the system, and one is talking about facts of the surrounding logic.

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