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Impossible to have solution to the equation. Induction Proof Problem

  1. Mar 16, 2013 #1
    1. The problem statement, all variables and given/known data

    For all integers n, it is impossible to have a solution to the equation

    4^n = a^2 + b^2 + c^2

    where a, b and c are all positive integers. (Hint: Notice that 4^n = 2^2n is a perfect square. Show (prove) that if m^2 = a^2 + b^2 + c^2, then we must have that a, b and c are all even. This can be done without induction; just think about what remainders a perfect square can leave when divided by four.)


    2. Relevant equations

    None in accordance from the chapter. I'm in a discrete mathematics course and this problem's chapter is on Induction. With the previous chapter before it was dealing with standard logic proofs.

    3. The attempt at a solution

    The only idea I could think of to attempt with is using the proof method of induction just like how I have used in the previous problems of my homework. By induction proof method, I mean the procedure of plugging in "n=1", to see if the proposition is true and continuing after by plugging in "n+1" etc. I'm just not sure how to adjust this problem to solve it by using induction.
     
  2. jcsd
  3. Mar 16, 2013 #2

    Dick

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    Homework Helper

    The way you want to solve it is not quite the same as that form of induction. If n=0 then there is no solution. It's easy to check for n=1 there is also no solution. So if there is one, there must be a SMALLEST value of n that gives a solution. If you've figured out why a, b and c must all be even then you should be able to show that there is also a solution for a SMALLER value of n. That would be a logical contradiction of n being the SMALLEST. Hence there is no solution. Can you fill in the details?
     
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