A function for all you math nerds out there to analyze
Dustan
I've already done a lot on this, but instead of giving away my
analysis, I'd like to see what other people come up with. I promise
that if no one covers it all, I will eventually put forward my
findings as well. I am interested in particular in integer solutions,
and what the graph as a whole looks like.
jank...@hotmail.com
This is a pretty well known problem. See, for example:
http://www.qbyte.org/puzzles/p048s.html
---
J K Haugland
http://home.no.net/zamunda
William Elliot
Typical nerd. Are x and y, to use conventional notation, reals
and not complex numbers? By any chance are they positive reals?
Clearly the solution set includes the diagonal. But what diagonal?
Are these solutions: x = y = 0, x = y = -1, x = y = -1/2; x = y = i?
Thus uncountably many solutions.
Ok, there you are, an opportunity to state
clearly and precisely the exact problem.
----
Dustan
> Typical nerd.
Thanks! Anyway, I could take the time to dissect every part of this
post, but instead, I'm just going to point out that my original post
was intended to give free reign. Please, any analysis goes.
N. Silver
As J K Haugland has pointed out, there are a lot of references.
If we restrict ourselves to positive integer solutions, we get:
(x = 2 and y = 4) or (x = 4 and y = 2).
We can show this by taking logs:
log x^y = log y^x.
Then we get y log x = x log y
and (log x)/x = (log y)/y.
We analyze the function f(t) = (log t)/t.
If we use the natural logarithm, then f
has the maximum value of 1/e at t = e,
and f increases on the interval 0 < t < e
and decreases on e < t < infinity.
To solve our problem, we need one
value of t less than e, say t = x, and one
value of t greater than e, say t = y, with
the property (log x)/x = (log y)/y.
But the only candidates for x are the two
positive integers less than e, namely 1 and 2.
David W. Cantrell
The problem with x and y restricted to being positive reals is well known,
and that's what your link seems to concern. But the problem without that
restriction is not so well known. If Dustan is particularly interested in
integer solutions, then we also have solutions (x,y) such as (-1,-1) and
(-4,-2).
In the thread "solve x^y = y^x" (sci.math, Apr. 2000), I state all
solutions for which x, y and x^y are real, and I describe what the
corresponding graph looks like.
David W. Cantrell
<http://groups.google.com/group/sci.math/msg/42d6e57bac1c90ef>
Michael Press
<76116025-d979-428c...@o10g2000hsf.googlegroups.com>,
Dustan <Dustan...@gmail.com> wrote:
^^^^^
rein
You seriously underestimate the mathematical abilities
of the folks with whom you argue. You can discount my
abilities as much as you like without risk of being
shown to have underestimated them.
--
Michael Press
Dustan
> You seriously underestimate the mathematical abilities
> of the folks with whom you argue.
How can I underestimate what I never estimated?
William tried to make a fool of me by suggesting that perhaps I was
limiting 'analysis' to positive reals, when in fact, I have already
accounted fully for positive reals by myself, and, in fact, WANTED
them to go outside those bounds. I ANTICIPATED that they would be able
to find out more about the other 3 quadrants I was missing on the real
number graph, which I got thanks to David W. Cantrell, but in
addition, I got someone who thought I was, and criticized me for,
asking specifically for positive reals! And then I get someone
criticizing me for God knows what; I can guess that you also thought I
was limiting to positive reals, but it's only a guess.
If this is the kind of response I should expect whenever I put forward
something that I find interesting on this newsgroup, instead of a
healthy discussion of the prompt, then I must ask: why am I posting
here? I could just as easily have remained ignorant of 99.9% of the
analysis that's been done on the relation X^Y = Y^X. That would
certainly have certainly been a bummer for me, but then I wouldn't
have had to suffer the abuse of snobby college professors (or whoever
you and William are) who think I'm only interested in a tiny subset of
all there is to understand.
It seems to me that, while you say that I have underestimated YOUR
abilities, YOU have underestimated the extent to which I am willing to
learn about this particular relation, having spent an afternoon
finding out about the first quadrant, inventing my own parametric form
of the relation, finding out what it looks like, demonstrating (and
proving) that the intersection between the hyperbolic-like shape and
the y=x line is at (e, e), etc, etc. I tried to be as clear as
possible that I wanted a completely free-reign discussion of all
aspects of this relation that one might study, but two people so far
have only spent their time criticizing my prompt, and not addressing
the prompt itself.
The fact that you were unable to take the hint and realize that I
really did intend a completely free-reign discussion, not one limited
to positive reals, or even reals, for that matter, really says
something about your ability to read and take a hint. I warned you in
my response to William Elliot that I could just as easily have torn
his condescending post to pieces, and instead chose to keep a civil
tongue and repeat my prompt in a perhaps clearer form. If you're going
to respond telling me there was no need to rant, don't bother. You
brought it upon yourself. I, meanwhile, will continue to be helpful
where I can on this newsgroup, because it is clear to me that such
kind, helpful people are desperately needed here to cover up for the
rest of the mathematicians.