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Topic: x^x is irrational (Read 3271 times) |
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NickH
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x^x is irrational
« on: Sep 7th, 2002, 3:16am » |
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If x is a positive rational number, prove that x^x is irrational unless x is an integer.
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| « Last Edit: Oct 23rd, 2003, 7:44pm by Icarus » |
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rugga
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Re: NEW PROBLEM: x^x
« Reply #1 on: Sep 7th, 2002, 2:09pm » |
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Nice problem, NickH. How's this?
"x is a positive rational number" means x = a/b for some relatively prime positive integers a & b. If xx is rational, then it can be written as c/d for some relatively prime integers c & d. So
c/d = xx = (a/b)a/b
= ((a/b)a)1/b
= ( aa / ba ) 1/b
Raising both sides to the power of b:
cb / db = aa / ba
or
cb ba = aa db
Assume for a moment that b>1. Then choose some prime factor of b (call it p). Consider the number of times p occurs in the prime factorization of each of the above terms:
ba: ka, where k is the number of times p occurs in b
aa: 0, since a and b are relatively prime
db: jb, where j is the number of times p occurs in d (Since p occurs on the left-hand-side, it must also occur on the right-hand-side, so it must be a factor of d.)
cb: 0, since c and d are relatively prime
So ka=jb since the number of occurrences has to match up on both sides of the equation. But since b is relatively prime to a, b must divide k.
In other words, the number of times p occurs in b (namely k) is a multiple of b itself, which means b is a multiple of pb.
This is a contradiction since pb is greater than b for any integers b,p>1. Therefore b can't be greater than 1 if xx is rational.
Of course if b=1 then x is an integer for any integer a, and xx is rational.
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NickH
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Re: NEW PROBLEM: x^x
« Reply #2 on: Sep 7th, 2002, 2:53pm » |
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That's exactly the method I used!
The question came up as part of a proof that the solution of x^x = 2 is transcendental. The above result proves that x cannot be rational.
The second step was to use something called the Gelfond-Schneider Theorem. This states that if a is algebraic and not equal to 0 or 1, and b is algebraic and irrational, then a^b is transcendental. This proves that x cannot be algebraic and irrational, and therefore is transendental.
Nick
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