Irregular Webcomic! #1807

Pierre de Fermat was a French lawyer and an amateur mathematician, with a particular interest in number theory and diophantine equations (polynomial equations with variables constrained to integer values).

Anyone who has studied calculus for at least a month or two will be familiar with one of Fermat's many mathematical theorems. The one known simply as Fermat's theorem says that local maxima and minima of a differentiable function are stationary points; i.e. the value of the derivative at a maximum or minimum is zero. This is the underpinning of a standard technique in calculus, and used and loved by school students everywhere.

He is also known for a number of other theorems, including:

Fermat's little theorem: For every integer a and every prime number p, the number a^p-a is divisible by p.
Sums of two squares: Every odd prime number which is 1 greater than a multiple of 4 is equal to the sum of two square numbers; conversely every odd prime number which is not 1 greater than a multiple of 4 is not equal to the sum of two square numbers.

The interesting thing about Fermat is that, as an amateur mathematician, he almost always stated his results without providing any proof for them. Over the centuries since his death in 1665, this has caused various amounts of intrigue and grief to a number of later mathematicians. For example, Fermat's little theorem was first proved by Gottfried Leibniz, around 1683, nearly 20 years after Fermat had died.

Fermat's theorem on the sums of two squares remained unproven until the great Leonhard Euler finally found a proof for it, in 1749.

Fermat is probably best known, however, for what came to be known as Fermat's last theorem. This wasn't the last theorem he wrote (he wrote it in 1637), but it was the last one to be proven. In fact, for a long time, nobody was sure if it was actually true or not. A long time. Over 300 years.

It was proven, finally, in 1994.

It was proven thanks to the concerted efforts of a series of highly skilled professional mathematicians in a sequence of work dating back to the 1960s, which laid the theoretical foundations of what would turn into a monstrous multi-faceted edifice of inter-related work. These efforts slowly chipped away at various aspects of the theorem, showing that it was equivalent to other, more complicated statements of number theory that could themselves be attacked by other analytical methods. Eventually, in 1994, the last bricks were put into place and Fermat's last theorem was proven to be true.

The really interesting thing about Fermat's last theorem is what Fermat himself wrote about it. He had a copy of the 3rd century Greek mathematical text Arithmetica, by Diophantus (after whom diophantine equations are named). In this book, Diophantus poses a problem closely related to Pythagoras' theorem (the most popular theorem in all of mathematics, according to a reliable source). He asks how to solve equations of the form:

a² = b² + c².

In the margin next to this, in his copy of Arithmetica, Fermat wrote (in Latin):

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Translating the 17th century mathematical jargon into modern terms, this reads:

There are no integer solutions for the equation aⁿ = bⁿ + cⁿ for n > 2. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

This final throwaway line of Fermat's has gone down in the annals of history as one of the most famous comments ever written about mathematics. Taking it at face value, and aware of the fact that Fermat discovered and recorded many other powerful theorems that were later proven to be true, one could conclude that Fermat did indeed have some sort of proof for his statement.

Although we know now that he was right, the proof only came about because of a concerted effort by dozens of mathematicians using 20th century analytical techniques that would have been far beyond the understanding of mathematicians of the 17th century. This is clearly incompatible with whatever conjectured proof that Fermat may have had. If he had a proof, it would have been relatively simple and elegant - a supposition supported by his claim that his proof was "truly marvellous".

The failure of anyone else to find a simple and elegant proof of Fermat's last theorem for the past 370 years raises a couple of possibilities:

Fermat did indeed have a relatively simple proof, but nobody else has been clever enough to reproduce it.
Fermat was mistaken about his proof, but happened to pick some statement about mathematics that was in fact true, but would take mathematics nearly 400 years more advanced to actually prove.

Either of these possibilities is fairly mind-boggling. Of course, there's always the other choice:

Fermat was a time traveller.

Iustinus offers this translation of da Vinci's note into Latin:

Decipulam mortiferam sane detexui. Hanc musei exiguitas non caperet.

He notes that was constructed to keep the initial letters of all the words the same as the corresponding parts of Fermat's marginal note.

2018-08-24 Rerun commentary: That is a very... um... blocky looking piece of paper falling from the ceiling. I suppose I could have tried using a real piece of paper that was more flexible, but then I'd have had the problem of it being something that Minnesota Jones couldn't easily hold with his LEGO hand.

Leoardo did actually use paper, not other materials such as parchment (made from animal skin)^[1]. The paper was made from hemp or linen rags. He was born at the right time for this because during the 15th century the spread of the printing press led to a demand for large quantities of a material suitable for printing on, which in turn led to the widespread production of paper. If Leonardo had lived a few decades earlier, he probably would have made his drawings and notes on parchment instead.

[1] "Drawing materials used by Leonardo da Vinci", Royal Collection Trust, https://www.royalcollection.org.uk/sites/default/files/resources/Drawing%20materials%20Leonardo%202019.pdf.

Post scriptum: Leonardo was, of course, a time traveller, so would have been familiar with all sorts of writing materials. I have a truly marvellous proof of this fact, but this annotation is too small to contain it.