The Fermat Curve and The Golden Ratio

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"$\mathbb Q(sqrt{5})$ is the field having the least discriminant such that $a^3 + b^3 = c^3$" has a non-trivial solution."

I never found out if this indeed was the case, but I asked the teacher about finding a point on this Fermat curve for this exponent over this quadratic field. Much to my surprise

"Let $\phi$ be the Golden Ratio. If $a:b:c$ is equivalent to $4+\phi : 5-\phi : 6$, then $a^3 + b^3 = c^3$."

This reminds me of the similar statement "If $a:b:c:d$ is equivalent to $3:4:5:6$ then $a^3 + b^3 + c^3 = d^3$. Still, I thought this was too cute of an identity not to pass up!


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just call me Dhelia , whatever where you are , i'm coming for you

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