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There are many beautiful infinite radical equations, some relatively straightforward, some much more subtle: $$ x = \sqrt{ x \sqrt{ x \sqrt{ x \sqrt{ \cdots } } } } $$ $$ \sqrt{2} = \sqrt{ 2/2 + \sqrt{ 2/2^2 + \sqrt{ 2/2^4 + + \sqrt{ 2/2^8 + \sqrt{ \cdots}}}}} $$ $$ 3 = \sqrt{1 + 2\sqrt{1 + 3\sqrt{ 1 + 4\sqrt{ \cdots }}}} $$

But I have seen far fewer analogous equations for $n$-roots. Here is one: $$ 2 = \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{\cdots}}}} $$

My question is:

Q. Are there truly "more" beautiful infinite radical equations, or is it just our natural gravitation toward the simpler square-root equations that leads to collections emphasizing radicals?

I am aware this question is vague, but perhaps some nevertheless have insights.

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For any integer $n \ge 2$ we have:

$n = \sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\cdots}}}}}$.

This can be generalized for $m$-th roots:

$n = \sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\cdots}}}}}$.

This generates an infinite family of nested radical equations. Of course, this doesn't cover every nested radical equation out there. Now, how many equations in this infinite family are beautiful?

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  • $\begingroup$ Point taken! These are beautiful but simple, just solving $x=( n^m -n + x )^{1/m}$. It is natural to wonder if there are others as subtle as Ramanujan's formula for 3, my 3rd example. $\endgroup$ – Joseph O'Rourke Jul 2 '14 at 10:54
  • $\begingroup$ @JimmyK4542, what about $\sqrt[n]{\sqrt[n]{x}}$? $\endgroup$ – Buffer Over Read Dec 11 '16 at 20:26
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Here's a nested radical for 4th roots that is not so well known. Given the tetranacci numbers (an analogue of the fibonacci numbers). Let $y$ be the tetranacci constant, or the positive real root of,

$$y^4-y^3-y^2-y-1=0$$

Then,

$$y(3-y) = \sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-\dots}}}} = 2.06719\dots$$

The family can be found in this MSE post.

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