[tex] f(x) = x^x [/tex]Given this function, defined, let's say for

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Discussion Overview

The discussion revolves around the function \( f(x) = x^x \) and explores the conditions under which \( f(x) \) yields integer values for rational versus irrational inputs. Participants examine whether there are rational values of \( x \) that can produce integer outputs and the implications of these findings.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks if there is a method to determine when \( x \) is rational versus irrational for integer values of \( f(x) \).
  • Another participant suggests taking the xth root of \( f(x) \) as a potential approach.
  • A later reply clarifies that they meant the xth root of \( f(x) \) and expresses uncertainty about the problem.
  • One participant posits that if \( x^x \) is an integer, then \( x \) must be either an integer or transcendental, based on their exploration of rational inputs.
  • Another participant questions whether non-integer rational values, such as \( 1/3 \) or \( 1/4 \), could yield integer outputs, indicating a belief that only integer inputs produce integer outputs.
  • One participant mentions the possibility of using logarithms to analyze the function further.
  • A participant recalls finding an article stating that there is no known way to rearrange the equation \( y = x^x \) into a function of \( y \) in terms of \( x \), but mentions an iterative formula for approximating \( x \) given \( y \).

Areas of Agreement / Disagreement

Participants express uncertainty and explore various hypotheses regarding the relationship between rational and irrational inputs and integer outputs. There is no consensus on whether non-integer rational values can yield integer outputs, and multiple competing views remain regarding the nature of solutions.

Contextual Notes

Participants discuss the implications of assuming \( x \) is an integer and the challenges in determining the nature of \( x \) when \( f(x) \) is an integer. The discussion includes unresolved mathematical steps and assumptions about the properties of rational and irrational numbers.

Paul
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[tex]f(x) = x^x[/tex]
Given this function, defined, let's say for all real numbers, is there any way to tell when x is rational versus irrational for integer values of f(x)?
e.g.
[tex]x^x = 4[/tex]
x = 2
[tex]x^x = 27[/tex]
x = 3
[tex]x^x = 3[/tex]
x = 1.825455054...

Thanks!
 
Last edited:
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You mean other that simply taking the xth root of x?
 
Do you mean the xth root of f(x)? Or am I misunderstanding? And yes, I meant other than simply looking at specific argument and function values.
 
Paul said:
Do you mean the xth root of f(x)?

Whoops, I sure do. :redface:

And yes, I meant other than simply looking at specific argument and function values.

I'm not sure off the top of my head, but let me play with it.
 
3^3 doesn't equal, so x=3 is irrational. sorry to nitpick.

interesting problem though. i bet there isn't a way besides applying the interger set for x and assuming all others will be irrational. is there any value for f(x) that results in a non interger rational number? if so, that negates using the interger set to find the y.
 
Thanks, I corrected the typo. And I guess that would put us on the path to a solution. Essentially, given that x is not an element of Z (the integer set), is [tex]x^x \in Z[/tex] possible?
 
Okay, suppose x=a/b, where a,b are integers. Assume that the fraction is reduced, ie. gcd(a,b)=1. Then x^x= (a/b)^(a/b)=[tex](\frac{a^a}{b^a})^\frac{1}{b}[/tex]

But [tex]gcd(a^a,b^a)=1[/tex], and therefore [tex](\frac{a^a}{b^a})^\frac{1}{b}[/tex] is irrational. It follows that if x^x is an integer, then either x is an integer or transcendental.
 
so there are also no solutions that are non intergers but rational, like 1/3 and 1/4?
i understand that x^x is an interger if x is an interger.

that tells us that the set of intergers for x gives us the solution set of y? there are no fractional xs for interger ys? sorry I am dumb. i want to learn though!
 
I was wondering the same thing about taking the logarithm or ln of a function.
 
  • #10
I was interested in this quite some time ago. I finally found an article saying there is no known way of rearranging the equation:

[tex]y=x^x[/tex]

In to some function of y in terms of x. But I did find an iterative formulae so you could approximate to as much accuracy as you wanted if you had y and wanted to know x. I'll see if I can find it again.
 

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