# The Amateur Mathematician

Pierre de Fermat

Birth: 1601, France

Death: 1665, France

Two names lead the first half of 17th-century mathematics: Rene Descartes and Pierre de Fermat. It is rather unusual since neither of them saw themselves as mathematicians as first.

Pierre de Fermat worked as a lawyer and a government official from 1631 until his death. He is regarded as the most important amateur mathematician of all time.

Entry to math circles

Fermat started working on mathematics in the late 1620s. He loves creating problems in the number theory and proving them.

Fermat became famous among mathematicians with his letter to Mersenne* dated back to April 26th, 1636. In his letter, Fermat worked for subjects such as Galileo’s free-fall experiment and Apollonius’ conics which led him to correspond with many mathematicians.

Even though his famous letter involved mathematical physics, he was interested in number theory. This is why, whatever the subject was in his letters, he somehow brought it to a problem in number theory. He wanted mathematicians to prove the problems he created and solved. But these problems were incredibly hard, and soon mathematicians started irritated by him. For instance, Frenicle de Bessy thought that Fermat was teasing him with his difficult problems.

Last Theorem

Today, Fermat is best known for his famous last theorem: Fermat’s Last Theorem. Fermat didn’t consider himself as a mathematician. Hence, he didn’t publish any of his works. In fact, sometimes he was writing his theorems and proofs in the blank parts of books.

One of those theorems was Fermat’s Last Theorem*.

* Fermat’s Last Theorem

“No three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.”

Fermat also noted the following:

“I have discovered a truly remarkable proof which this margin is too small to contain.”

Fermat’s Last Theorem was finally proved by British mathematician Andrew Wiles in 1994. That means the theorem was unproven for almost 358 years!

Alienation

Between 1643 and 1654, Fermat was alienated by the science community. There were a few reasons for that such as the civil war and the plague that affected where he lived. But one of the main reasons was his dispute with Descartes.

In his time, Descartes was influential among French scientists. So, when Fermat made a negative comment about his beloved work La Geometrie, Descartes went after him. He (wrongly) criticized Fermat’s work for maxima-minima-tangent*.

Even after Fermat proved that his work was complete, Descartes continued the argument by claiming that Fermat wasn’t a sufficient mathematician.

*Maxima-Minima-Tangent

You probably heard derivatives and integral if you took a math course in university. Actually, the first developments of those two subjects belonged to a 17th-century French lawyer.

Using a simple example, I will show you how Fermat’s method worked:

Assume that we have a straight line with length a:

My ambition is to find a point on it such that it separates the line into two parts, and multiplication of the length of those two parts is maximum.

Assume that one of those parts has length x. Then the other part would have length a-x:

Then, the maximum becomes:

x . (a-x)

ax – x2

So, how can we find this length x?

At this point, Fermat discovered an ingenious method. He says that we should add length e to x:

But this length e should be so small, we might as well consider it as zero. In other words, e is infinitely small. Now we can substitute every x in our solution with x+e:

a(x+e) – (x+e)2

Use basic algebra to simplify the equation:

ax + ea – x2 – 2xe – e2

From this we can reach;

ax – x2 = ax + ea – x2 – 2xe – e2

Again simplify the equations:

ea = 2xe – e2

Now, divide everything by e:

a = 2x – e

Remember that Fermat said e is so close to zero, it is in fact zero? Then:

a = 2x

Our line a has length 2x. This means that if we split a straight line into two equal parts, their multiplication becomes maximum.

Final Years

After his dispute with Descartes ended, Fermat got in touch with mathematicians. His correspondence with Blaise Pascal led to the set up of the probability theory.

Fermat was far ahead of his time in number theory. He almost carried the whole branch by himself in the 17th-century. His findings and problems were appreciated much after his death in the modern number theory. For me, his title will always be the greatest amateur mathematician of all time.

M. Serkan Kalaycıoğlu