## logic verification

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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
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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