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Analysis - irrational number or positive integer

  1. Sep 12, 2007 #1
    1. The problem statement, all variables and given/known data

    Let n and k be positive integers. Show that [tex]k^{1/n}[/tex] is either a positive integer or an irrational number.

    3. The attempt at a solution

    I set [tex]q = k^{1/n}[/tex]. Then I set [tex]q = \frac{m}{p} [/tex]. (Where m and p don't have common factors.) Then [tex]m^n = k * p^n [/tex]. So then k is a factor of [tex]m^n[/tex].

    But here I get stuck. In other proofs they usually show that, like, then k must also be a factor of m, (but I don't know how to do that, if it is true), and then so [tex]m^n = [/tex] an integer * k^2, so then k must also be a factor of [tex]p^n[/tex], which means that m and p do have a common factor.

    But I get stuck in the middle.
     
    Last edited: Sep 12, 2007
  2. jcsd
  3. Sep 12, 2007 #2

    EnumaElish

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    What are you assuming about m and p, are they positive integers? If so, haven't you assumed q is rational?
     
  4. Sep 12, 2007 #3
    StatusX - Thanks so much for the help. If the factor divides m^n, how do I show that it also divides m?

    EnumaElish - I am assuming m and p are positive integers with no common factors. So yeah, I have assumed q is rational. I am trying to prove that it cannot be rational (unless k is an nth power).
     
  5. Sep 12, 2007 #4
    I think you are close but going about it the wrong way. The easiest way to prove things like this is to assume the opposite.

    Suppose q isn’t irrational or an integer.

    This means that q can be written in the form of a/b where a and b are relatively prime integers and b is non zero and non 1.

    So q = k^(1/n) = a/b.

    But this means that k = a^n/b^n

    But what did we assume? That a and b are relatively prime, and clearly if a and b are relatively prime so are a^n and b^n. But this means that a^n/b^n isn’t an integer. But we assumed that k is an integer. This leads us to a contradiction, hence our assumption is false. QED.

    Now since I gave you a pretty solid outline of the proof, a good way to make sure you really understand it would be to go back and see if you can spot where I used every assumption. Where did I use the fact that a,b and n are integers, where did i use b isn't 1 or 0? etc.
     
  6. Sep 12, 2007 #5

    StatusX

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    Sorry, I deleted that post because it had a mistake. Just use the prime itself. That is, assume q is a prime factor of k whose power in k is not a multiple of n. Then q must divide m^n, and so clearly also m.
     
  7. Sep 12, 2007 #6
    JohnF - Thanks, I think I understand better. That proof makes sense to me.

    StatusX - I don't understand how q must divide m if it divides m^n.
     
  8. Sep 12, 2007 #7

    StatusX

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    Because it's a prime. Just think about it a little.
     
  9. Sep 14, 2007 #8

    Gib Z

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    There seems to be more conditions required (q=4, m=n=2)
     
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