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logic verification

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  • David Litchfield
    I know FLT has been proved for all cases but specifically for the power being 3 does this logic follow? Cheers, David Assume that a^3 + b^3 = c^3 where a 1
    Message 1 of 1 , Jan 5, 2003
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      I know FLT has been proved for all cases but specifically for the power
      being 3 does this logic follow?
      Cheers,
      David

      Assume that a^3 + b^3 = c^3 where a > 1 and b > 1

      By factoring a^3 + b^3 we know that a^3 + b^3 = (b+a)(a^2 + b^2 - ba)

      so a^3 + b^3 = c^3 = (b+a)(a^2 + b^2 - ba)

      So for a^3 + b^3 to equal c^3 then (a^2 + b^2 - ba) must be equivalent to
      (b+a)^2 so that (b+a)(b+a)^2 = (b+a)^3

      Expanding (b+a)^2 we get

      (b+a)(b+a) = a^2 +b^2 + 2ba

      Replacing (b+a)(b+a) with the original a^2 + b^2 - ba we get

      a^2 + b^2 - ba = a^2 +b^2 + 2ba

      Simplifying this we get

      -ba = 2ba

      3ba = 0

      If 3ba = 0 then one of b or a must be 0 but as both b and a are greater than
      1 then we are left with a contradiction.

      So a^3 + b^3 cannot equal c^3
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