Wikipedia:Reference desk/Archives/Mathematics/2017 September 5

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September 5

Floor function theorem

How to prove that the amount of integers between $1,\ldots ,n$ which are divided by $1\leq k\leq n$ is $\left\lfloor {\frac {n}{k}}\right\rfloor$ ? יהודה שמחה ולדמן (talk) 19:09, 5 September 2017 (UTC)

Is not that obvious? Ruslik_Zero 20:35, 5 September 2017 (UTC)

Hint: Show that $k\left\lfloor {\frac {n}{k}}\right\rfloor$ is such a number, and that it is the largest one. -- Meni Rosenfeld (talk) 20:56, 5 September 2017 (UTC)

No it's not obvious to me! And please don't go from the end to the beginning.

Show me with the function's definitions and properties. יהודה שמחה ולדמן (talk) 21:00, 5 September 2017 (UTC)

Is this a good proof?

By the Euclidean algorithm $n=qk+r$ for some $q,r\in \mathbb {Z} \ ,\ 0\leq r<k$ .

By properties we get

$\left\lfloor {\frac {qk+r}{k}}\right\rfloor =\left\lfloor q+{\frac {r}{k}}\right\rfloor =q+{\color {blue}\left\lfloor {\frac {r}{k}}\right\rfloor }=q+\color {blue}{0}$

But is there another way to prove this? I think this is a "$\Leftarrow$ proof" (from the way it starts) and I want a "$\Rightarrow$ proof". יהודה שמחה ולדמן (talk) 22:47, 5 September 2017 (UTC)

By the definition of divisibility, the integers $m$ divisible by $k$ satisfy $m=qk,q\in \mathbb {Z}$. We are given $1\leq m\leq n\Rightarrow 1\leq qk\leq n\Rightarrow 1\leq q\leq \left\lfloor {\frac {n}{k}}\right\rfloor$. The last implication follows from the definitions of inequality, division, and floor. C0617470r (talk) 07:47, 6 September 2017 (UTC)

Sorry, didn't get the last step. יהודה שמחה ולדמן (talk) 12:51, 6 September 2017 (UTC)

$\left\lfloor {\frac {n}{k}}\right\rfloor$ is by definition the largest integer which is $\leq {\frac {n}{k}}$. $q$ is an integer which is $\leq {\frac {n}{k}}$. What does that say about the relation between $q$ and $\left\lfloor {\frac {n}{k}}\right\rfloor$? -- Meni Rosenfeld (talk) 15:25, 6 September 2017 (UTC)

We get $0\leq \left\lfloor {\frac {1}{k}}\right\rfloor \leq 1\leq q\leq \left\lfloor {\frac {n}{k}}\right\rfloor \leq {\frac {n}{k}}\leq n$ . But how does it prove $q=\left\lfloor {\frac {n}{k}}\right\rfloor$ ? יהודה שמחה ולדמן (talk) 16:03, 6 September 2017 (UTC)

Oh! I think I get it now! How could I have ben so blind!?

It's all in the definition of $\left\lfloor {\frac {n}{k}}\right\rfloor \leq {\frac {n}{k}}$ (and "maybe" $n=qk+r$ too). יהודה שמחה ולדמן (talk) 16:48, 6 September 2017 (UTC)

LaTeX mathematics color code

Why does \color{"color"}{"formula"} work on wikipedia, but on wikibooks you must use {\color{"color"}"formula"}? יהודה שמחה ולדמן (talk) 19:35, 5 September 2017 (UTC)

Please post a real example. I tried <math>\color{red}{42}</math> and it worked both at the English Wikibooks and here: $\color {red}{42}$. PrimeHunter (talk) 21:32, 5 September 2017 (UTC)

Forgot to mention – \color{"color"}{"formula"} works on wikibooks, but it colors all the characters outside the brackets too. יהודה שמחה ולדמן (talk) 21:38, 5 September 2017 (UTC)

PLEASE POST A REAL EXAMPLE. Not pseudocode or a description but an actual example which renders differently at Wikipedia and Wikibooks. What is your math setting at Special:Preferences#mw-prefsection-rendering at Wikipedia and Wikibooks? Are you using the same browser? There have been cases where different Wikimedia servers gave different results for formulas. PrimeHunter (talk) 22:08, 5 September 2017 (UTC)

I'm using PNG images at both. יהודה שמחה ולדמן (talk) 22:12, 5 September 2017 (UTC)

Without an example of the alleged issue I give up trying to help. PrimeHunter (talk) 22:19, 5 September 2017 (UTC)

Here: $\color {blue}{a}+b$ . יהודה שמחה ולדמן (talk) 22:36, 5 September 2017 (UTC)

It's fully blue for me at both Wikipedia and Wikibooks. Is "+ b" black for you here? Help:Displaying a formula#Color mentions there can be a difference for some notations. PrimeHunter (talk) 23:23, 5 September 2017 (UTC)

Fully blue for me. יהודה שמחה ולדמן (talk) 23:33, 5 September 2017 (UTC)

So do you have an example with the alleged difference between Wikipedia and Wikibooks? If your post is just about invalid color code not giving the result you want then change it to valid code per Help:Displaying a formula#Color. PrimeHunter (talk) 12:54, 6 September 2017 (UTC)

Are you saying I could redefine LaTeX codes? יהודה שמחה ולדמן (talk) 13:05, 6 September 2017 (UTC)

I tried your example in two external online LaTeX editors and both rendered it fully blue. I'm not a LaTeX expert but I think you just have a wrong expectation about where the color ends when you don't specify it with a bracket around \color. Anyway, Help:Displaying a formula says: "MediaWiki renders mathematical equations using a combination of html markup and a variant of LaTeX." Do you have example code which doesn't render as MediaWiki documentation says it should? If so then please give the exact code and say where you tested it and how it differs from your expectation. PrimeHunter (talk) 13:52, 6 September 2017 (UTC)

@יהודה שמחה ולדמן: What PrimeHunter is trying to get out of you is something in the form "I did X, I expected Y, but Z≠Y happened", with X, Y and Z very precisely stated. A very important reason for that is that you may answer your own question when trying to produce such a description. I encourage you to read [1] for more details. Tigraan^{Click here to contact me} 14:37, 6 September 2017 (UTC)

integer points on a particular elliptic curve

The equation is y^2=x^3-x+3. Are there any integral solutions besides (2,3) and (2,-3)?144.35.45.46 (talk) 22:55, 5 September 2017 (UTC)

Nope. You can find integral solutions using Sage or Mathematica. C0617470r (talk) 07:20, 6 September 2017 (UTC)