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Elementary math proof

  1. Apr 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Disprove the following: There exists a polynomial f(x) with integer coefficients such that f(1) is even and f(3) is odd.

    2. Relevant equations

    3. The attempt at a solution
    It's a little bit intuitive.
    1 and 3 have the same parity. They are both odd
    so if(odd)=odd then f(1)=odd and f(3)=odd
    or if(odd)=even then f(1)=even and f(3)=even

    is that right?
  2. jcsd
  3. Apr 30, 2016 #2

    Ray Vickson

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    No, it is not correct, because you are essentially assuming what you want to prove. In order to be able to assert that f(odd) = odd, for example, you need to show that it is not possible to have f(odd1) = odd but f(odd2) = even, etc.
  4. Apr 30, 2016 #3
    Hint: if there was such a polynomial ##f(3)-f(1)## would be both even and odd.
  5. Apr 30, 2016 #4
    why would they both be even and odd?
  6. Apr 30, 2016 #5


    Staff: Mentor

    What do you know about ##f(3) - f(1)##?
  7. May 1, 2016 #6


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    Let the polynomial be Σpnxn. What does f(3)-f(1) look like?
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