The Dictionary of Mathematical Eponymy: The Fermat Cubic

So far in the Dictionary of Mathematical Eponymy, I’ve not picked anyone properly famous. I mean, if you’re a keen recreational mathematician, you’ll have heard of Collatz or Banach; a serious mathematician might know about Daubechies, and a chess enthusiast would conceivably have come across Elo.

But everyone has heard of Fermat’s Last Theorem - which we’re not going to do. Instead, we’re going to look at Fermat’s Cubic.

What is it?

The Fermat cubic is a surface defined by the equation:

$$x^3 + y^3 + z^3 = 1$$

As soon as I saw it, this became my new favourite cubic surface. (I’m not sure I had one before). Just like Fermat’s Last Theorem is an obvious extension of Pythagoras to higher powers, this is the obvious extension of the sphere.

And just like Fermat’s Last Theorem is a somewhat more difficult problem than Pythagoras’s, the surface is a bit more wrinkly than the sphere.

I’m not going to go into its projective parametrisation here - but I can point you at a link if you’re interested.

Why is it important?

Expressing Fermat’s cubic parametrically is a classical Diophantine problem - it has a few well-known rational solutions, as Elkies mentions above.

The interesting thing for me is that Fermat’s cubic (or a curve closely related to it) is a key component in the current state-of-the-art in searching for solutions to the sums of three cubes problem - Andrew Booker recently found a solution to $x^3 + y^3 + z^3 = 33$. (Booker’s paper is here).

Who was Fermat?

Amusingly, Wikipedia refers to Pierre de Fermat as a ‘French lawyer’ - which isn’t inaccurate, but it’s like calling the late Sir Roger Bannister an ‘English doctor’. While lawyering presumably paid his bills, Fermat spent his spare time pushing the frontiers of mathematics.

Born in late 1607, Fermat is known for… well, his last theorem, obviously, but also a little one - Fermat’s Little Theorem states that $a^p \equiv a \pmod{p}$ (where $p$ is prime and $a$ is an integer). He laid many of the foundations for cartesian geometry and for the calculus, and made great leaps in number theory - among many other things named for him, he figured out how to factorise numbers using the difference of two squares, and studied numbers of the form $2^{2^n}+1$.

As if that weren’t plenty, he’s also one of the first mathematicians to make any headway with probability - and in physics, the idea that light takes the path that minimises its travel time is called Fermat’s principle.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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