Testwiki:Reference desk/Archives/Mathematics/2024 September 30

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September 30

How to find max(x^y) for each r where x + y = r and x, y are positive integers

Drawing fractal canopy diagrams had me wondering for a given SVG file size, I could add more branches or more depth. Below are two examples:

2 branches, depth 12

4 branches, depth 8

To make the tree as dense as possible, I wish to maximise the number of leaf nodes. In other words, how can I find max(x^y) for each r where x + y = r and x, y are positive integers?

I realise I can use calculus but I'm unsure what equation I should differentiate.

Thanks, cmɢʟee⎆τaʟκ 00:50, 30 September 2024 (UTC)

Unfortunately, this is actually quite complicated. We can rewrite the problem as $\max (x^{r-x})$, and consider all real $x$ instead of just integers. Notice that $\ln (x^{r-x})=(r-x)\ln (x)$. Because $\ln$ is monotonic over $x$, $x^{r-x}$ is maximized when $(r-x)\ln (x)$ is. Its derivative is $-\ln (x)+(r-x)/x$, which is $0$ when $r=x+x\ln (x)$. This is a nice closed-form expression, but it's for $r$ in terms of $x$. Inverting it is complicated, but Wolfram Alpha gives us $x=r/W(er)$ where $W$ is the Lambert W function. While this is sort of a closed-form expression, it's still unfortunately annoying to work with since $W$ is implicitly defined as $W(z)=w\iff z=w{e}^w$. In any case, $x=r/W(er)$ is irrational for all positive rational $r\ne 1$. The proof of this is annoying so I've put it below, but ultimately what it implies is that when $r>1$, as far as I can tell, the best you can do is either round $x$ for convenience, or take floor/ceiling of $x$ and compare values to get the max over integers.

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1. Suppose $r\in {\mathbb{Q}}^{+}\mathrm{\setminus }\{1\}$, and let $w:=W(er)$.
2. $W(er)=0\Rightarrow r=0$ and $W(er)=1\Rightarrow r=1$, so $w\ne 0,1$ as well.
3. By definition, $er=w{e}^w$. Since $w\ne 0$, we can rearrange to get $r/w=e^{w-1}$.
4. Lindemann's 1882 theorem implies that if $k$ is nonzero rational, then $e^k$ is not only irrational, but transcendental as well.
5. If $w$ is rational, then since $w\ne 1$, $e^{w-1}$ is transcendental, while $r/w$ is rational, which is contradictory.
6. $w$ must be irrational and is nonzero, so $r/w$ is irrational.
7. We conclude that $r\in {\mathbb{Q}}^{+}\mathrm{\setminus }\{1\}$ implies $r/W(er)$ is irrational.

GalacticShoe (talk) 02:41, 30 September 2024 (UTC)

Here is a numerical recipe (Newton's method) for solving $r=x+x\ln x$ for real-valued $x$:

1. Set $x=9$
2. Iterate the replacement $x=x^{*}$ until convergence, where

   $x^{*}=\frac{x+r}{2+\ln x}.$

In practice ($r<90$), two iterations will bring you close enough; then test $\lfloor x\rfloor$ and $\lceil x\rceil$ to get the optimal integer value.  --Lambiam 10:17, 30 September 2024 (UTC)

Thanks so much, @GalacticShoe and @Lambiam. I thought analytically I was heading into a dead end. Guess solving it iteratively is still the best for small values. cmɢʟee⎆τaʟκ 11:45, 30 September 2024 (UTC)

P.S. It seems there's an interesting trend:

x=1 for r=2	(1 term):	1¹.
x=2 for r=3 to 4	(2 terms):	2¹ and 2².
x=3 for r=5 to 7	(3 terms):	3², 3³ and 3⁴.
x=4 for r=8 to 11	(4 terms):	4⁴, 4⁵, 4⁶ and 4⁷.

x changes when r is the x–1th triangular number + 2. Serendipity? cmɢʟee⎆τaʟκ 17:06, 30 September 2024 (UTC)

Unfortunately, serendipity. The number of $r$ for each $x$ is the sequence OEIS:A108414. Although it starts with $1,2,3,4$, it quickly levels out. The inverse triangular number function you're looking for is $\lceil (\sqrt{8r-7}-1)/2\rceil \approx \sqrt{2r}$, while $x=r/W(er)$ grows faster than $r/\ln (r)$, which in turn grows faster than $\sqrt{2r}$, hence the number of terms slowing down in growth. GalacticShoe (talk) 03:46, 1 October 2024 (UTC)

Note that, unlike for Template:Italics correction, these first differences in the r-values for which x changes do not only always increase but may even decrease.  --Lambiam 03:58, 1 October 2024 (UTC)

After some testing, it seems that rounding suffices at least for r < 100, possibly more. To simplify it, applying Newton's method twice, we can get this approximation which maximizes $x^y$ for these values of $x+y=r$:

$x=round\left(\frac{9+5.197225r}{2.373852+4.197225\ln (9+r)}\right)$

GalacticShoe (talk) 04:14, 1 October 2024 (UTC)

Thanks so much, @Lambiam and @GalacticShoe. I much appreciate the time and effort you've put into it. Cheers, cmɢʟee⎆τaʟκ 20:17, 2 October 2024 (UTC)