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@maartengm

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@stevenstrogatz

(the first is ...=(x+4)^4 of course, the solution is correct, just the tweet isn't)

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I'm wondering if this class of equations has a limiting root as n tends to infinity, and if so, what that limiting root is (where we take the sum of n consecutive n'th powers on the left hand side, as you did for n =4 and 5).

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On the other hand, it seems (intuitively) like the root should tend to infinity as n tends to infinity. Curious to see more examples and then a proof, however this all turns out! Seems like just a few minutes of work. Have fun!

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It's a bit more non-linear than I expected These are the intersections for powers 3..13 The distance in x is not constant.

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Actually, the x's for which the roots are hit grow at a constant pace. There's just this niggly bit at the beginning...

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(These are the roots for n=3..50) Probably there's a trivial thing lurking here when the algebra is worked out..

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Final tweet sorry, these are the differences between consecutive roots of the cases as N grows. The difference between roots converges, quickly. (so the slope converges to a mysterious number close to (0.44))

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This is interesting, Maarten! I bet you're right; this might be something we could figure out (maybe easily?) once we do the algebra.

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I tried briefly, but my mathematics are lacking.. The terms explode into large integers. I hope someone with more skill can get something out of it. From a hand-waving high-level perspective the conversion of the slope makes me think of a 1/n! term vanishing.

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I'll think about it after work today. Gotta go now! Thanks for the fun experiments.

Maarten Mortier

@maartengm

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@stevenstrogatz

Steven I was quite awake last night because of this problem. I wrote it down nicely here gist.github.com/buddhabrot/2e3 Could you reach out to anyone who can help with demystifying this? It's a little scary because the constant factor does not seem to make sense. cc

@johncarlosbaez

gist.github.com

polfamily.ipynb

GitHub Gist: instantly share code, notes, and snippets.

8:03 AM · Oct 22, 2021

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I am playing myself during piano lesson of son. I think it might you constant might be 1/ln(2)-1. I am sitting in car scribbling. Will write it out later...