Re:Tell HN: 2^4 == 4^2, extended to rationals
by AnimalMuppet on Saturday April 22, 2023 @02:45PM
(#35669688)
No, I did not prove that this was all the cases - not even that this was all the rational cases. I tried, years ago (I literally made this post by trying to recall what I had found over 40 years ago), but I did not find a proof that this was all the possibilities.
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Re:Tell HN: 2^4 == 4^2, extended to rationals
by AnimalMuppet on Sunday April 23, 2023 @07:25AM
(#35675298)
After some time to think, yes, I'm pretty sure that it's all solutions. The shape of the function in the article means that there is only one b for any a. The formula I gave gives a solution, so it must be the solution. If you can find an n to generate an a, the formula must give you the correct b.
(As n goes from 0 to infinity, a goes from 1 to e, so the formula covers the entire range of possibilities.)
In my formula, however, n may be irrational, even for some rational a. I haven't proven that.