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Proof about m/nth root of a prime.

  1. Jul 17, 2012 #1
    Lets take a prime number and raise it to m/n where m and n are coprime. x,y are coprime
    and I want to show that this is irrational.
    Proof: lets assume for the sake of contradiction that
    [itex] P^{\frac{m}{n}}=\frac{x}{y} [/itex]
    P is prime and m,n,x,y are integers.
    no we take both sides to the nth power and then multiply the y term over.
    [itex] y^n P^m=x^n [/itex]
    now we factor y and x into their prime factorization.
    [itex] {p^a.....{P_{t}}^b}^n P^m={{P_{q}}^c........}^n [/itex]
    okay so if the left side is equal to the right side.
    on the left side we know that P has at least m factors, but if P is also contained
    in y then it has m+n factors. but if x and y are coprime then x has no common factors with y so y cant have factors of P in it. so now if x contains multiples of P , in order to have the same amount of factors it must be true that ne=m where e is the number of factors of P in x.
    but this would imply that m and n are not coprime therefore this is a contradiction and our original number is irrational.
     
  2. jcsd
  3. Jul 17, 2012 #2

    chiro

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    Hey cragar.

    The thing I see is that when you get Pm = xn/yn, then it means that (x/y) have to be a prime since primes have no factors other than 1 or itself. But the only way for this to happen is if x and y are not co-prime and also relevant powers of that prime.

    Thus you have a contradiction and you have proved the result that there exists no rational form given x and y are co-prime. Also because we are dealing with a prime, we know that the RHS of the above must be an integer and also that it has a particular decomposition since primes only have factors of 1 and itself.

    Edit: Correction should have been / instead of *
     
    Last edited: Jul 17, 2012
  4. Jul 17, 2012 #3
    how did you get [itex] p^m=x^ny^n [/itex]
     
  5. Jul 17, 2012 #4

    chiro

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    Sorry, I've changed it: should have been / instead of *
     
  6. Jul 18, 2012 #5
    okay I see, ya that is also a contradiction too and right off the bat.
     
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