By way of contradiction, suppose that that there exists a solution to x^3 + y^3 = z^3. Of all such
solutions, choose one in which z is as small as possible. Call this solution x_0, y_0, z_0.

Without a loss of generality x_0 y_0 and gcd(x_0, y_0) = 1. This implies gcd(x_0, y_0, z_0) = 1.

Prove that gcd(x_0, y_0) = 1

I need this for a proof of Fermat's Last Theorem for when n=3. Thanks!

Question

By way of contradiction, suppose that that there exists a solution to x^3 + y^3 = z^3. Of all such
solutions, choose one in which z is as small as possible. Call this solution x_0, y_0, z_0.

Without a loss of generality x_0 > y_0 and gcd(x_0, y_0) = 1. This implies gcd(x_0, y_0, z_0) = 1.

Prove that gcd(x_0, y_0) = 1

I need this for a proof of Fermat's Last Theorem for when n=3. Thanks!

anonymous · Answer

I need to figure out the proof for this part:
Prove that gcd(x_0, y_0) = 1

anonymous · Answer

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