Re:Tell HN: 2^4 == 4^2, extended to rationals
by roninghost on Saturday April 22, 2023 @02:04PM
(#35669321)
There are infinete number of rationals that satisfy the condition.
The integer uniquenes proof relied on the shape of the function between 1 and e non inclusive, and 2 is the only integer in the open interval.
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.
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.
Did you prove that this are all the case? Or this are only the cases you found and the general case is still open?