I’ve been looking at the value *N(n)* of *N* that satisfies the equation

Thus turns out to be

where the O(1/*n*) term is about 1/400*n* for *n*>10.

I’ve verified this by calculation up to about *n*=1000, using Lenstra’s long integer package LIP.

This result is so beautiful and simple that it must be possible to prove it without brute-force calculation. If anyone has any suggestions as to how to begin then I’d be very grateful.

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Hi!

In your equation “Sigma from 1-n (N – i)^n = N^n”

What is “i”?

Thanks for your interesting web site,

Best Wishes,

Andrew

“i” was the bound variable in the summation. I’ve put it in explicitly now: thank you for pointing out the deficiency!